This article The German Government Is Paying For Refugees To Return Home For 3-Week Vacation on noted blog Zerohedge.com, lifted in its entirety from another source, claims:

I am in Germany on business and did not see the place overrun with refugees as on my last trip. So I made some inquiries. To my complete astonishment, the German government is actually giving refugees three weeks paid vacations INCLUDING airfare BACK to the very countries that claim they are fleeing because it is unsafe.

So in other words, despite claiming their lives would be at risk if they were forced to return home, the government is paying them for a vacation to the very place they claim to be fleeing. You just cannot make up such a completely insane government policy. I know someone who works with the refugees and they confirm they are on vacation back home. Therefore, asylum seekers are nonetheless returning to their homeland for a “short time”.

This seems thinly sourced. However, there is a German linked sourced article, but I don't speak that language.

Can someone confirm or deny that Germany is paying refugee vacation back to the countries from which the refugees are supposedly being persecuted?

So, just to be clear, I am just talking about the airfare being paid for, not hotels or other types of vacation expenses or accommodations, although if social welfare benefits are accruing and being paid during this time, that would be good information to have.

The blog post on ZeroHedge is a heavily distorted version of the original German article that is linked in it.

What the original article in "Die Welt" claims is that there are refugees that travel to the country they claim to be persecuted. That is the entirety of the actual claim from the article:

Asylberechtigte kehren zu Urlaubszwecken vorübergehend in jenes Land
zurück, aus dem sie offiziell geflüchtet sind.

asylees are returning temporarily for vacation back to the country they officially fled

The article states that the existence of such cases has been confirmed by German authorities. The article is not entirely clear on whether that applies to asylees returning temporarily, or if they actually confirm that the reason is a vacation.

The article never claims that they are paid for the vacation, this is probably a distortion of the following paragraph:

Generell gilt: Hartz-IV-Empfänger haben Anspruch auf 21 Tage
Ortsabwesenheit pro Jahr, ohne dass die Bezüge gesenkt werden.

recipients of Hartz IV (a specific type of social security) are entitled to 21 days of absence per year without reduction of payments.

So asylees can still get social security benefits while on vacation, which is nothing specific to them but applies to everyone that receives them.

There is another article about the topic in "Der Spiegel", which is slightly more skeptical about the claim. According to this article the authorities don't know if the purpose of any of them was vacation. They do have specific cases where e.g. asylees traveled back to make preparations for their planned return.

It is pretty clear that the claim that Germany is paying for vacations of refugees is a distortion of the fact that asylees receive social security in Germany.

The reality behind this claim is not that clear cut and easy to either confirm or dismiss with one word.

No

In principle, Germany does not pay refugees "to go on vacation in those countries they fled from."

That reads as if the German state actively encourages vacations into these countries and pays extra money just for the trip. It's more like: once a refugee is officially recognised, they are entitled to receive a small amount of money. That money will be paid out continually even if the refugee leaves his residence for the maximum amount of time he is allowed to. That maximum time is 21 days per year. But that makes this a:

Yes, technically, and in very few cases over all

Refugees in Germany get paid a minimum amount of money. Refugees have to stay were they are registered to receive these benefits (Residenzpflicht). They are however entitled to leave that place of residence for up to 21 days without cuts to these benefits. What they do in these circumstances is up them. They might go on "vacation" in another part of Germany, Europe or who knows where, maybe even their home country.

They have a limited amount of time for travel they are allowed to use. If they leave their residence, the government does not ask for what purpose. Therefore the government lacks any statistics that would allow to draw any reliable conclusions on this. Technically, it is a valid point to state that the German tax payer pays for such flights via these regular payments. But it should be emphasised that this in no way bound to any destination, homeland or not, and that these payments are quite parsimonious. Paying for a flight to Syria for example from that amount is difficult, to say the least.

There are continually cases like the scandalous one publicised.
One example is from 1998:

Dem 28-jährigen Iraker Ismat M. gewährten die deutschen Behörden im August 1997 Abschiebeschutz. Vier Monate später flog er freiwillig in seinen „Verfolgerstaat“ und blieb dort 108 Tage. Inzwischen sind auch Ismats Ehefrau und drei Kinder nach Deutschland eingereist und haben im April Asyl beantragt.

__Translation: The 28-year-old Iraqi Ismat M. was granted deportation protection by the German authorities in August 1997. Four months later he flew voluntarily to his "persecuting state" and stayed there for 108 days. In the meantime, Ismat's wife and three children have also entered Germany and applied for asylum in April.

Source: Herbert Reinke-Nobbe: "Asylanten auf Heimaturlaub", Focus, Magazin | Nr. 21 (1998).

In that article a number is given

„Bislang haben wir bei weit über 100 Irakern den Asylstatus beziehungsweise den Abschiebeschutz widerrufen.“

__Translation: "To date, we have revoked the asylum status or deportation protection of well over 100 Iraqis."

But that is of course not really helpful, as it is not clear whether all of this number had their status revoked for the same reasons or not even what timeframe is referred to.

The main point here is that the original article the claim is based on is from 2016 and saying this:

Asylberechtigte kehren zu Urlaubszwecken vorübergehend in jenes Land zurück, aus dem sie offiziell geflüchtet sind.
Die Bundesagentur für Arbeit bestätigte: „Es gibt solche Fälle.“ Offizielle Erhebungen lägen dazu aber nicht vor.
Generell gilt: Hartz-IV-Empfänger haben Anspruch auf 21 Tage Ortsabwesenheit pro Jahr, ohne dass die Bezüge gesenkt werden. Ein solcher Urlaub wird im Regelfall genehmigt. Eine Pflicht, den Urlaubsort gegenüber dem Staat anzugeben, besteht bislang nicht.

__Translation: Persons entitled to asylum temporarily return to the country from which they have officially fled for holiday purposes.
The Federal Employment Agency confirmed: "There are such cases." However, there were no official surveys available.
In general, Hartz IV recipients are entitled to 21 days' absence per year without any reduction in pay. Such leave is usually approved. So far there is no obligation to state the holiday resort to the state.

Manuel Bewarder, Christoph B. Schiltz: "Flüchtlinge machen Urlaub, wo sie angeblich verfolgt werden", Welt, 11.09.2016.

That article goes on that a vacation in a persecuting country might be indicative that said state is not really persecuting the refugee and that government agencies were told to look into that matter and try to generate that needed data. They failed to do so in a timely manner. Given that there was no legal basis for such inquiries, even if refugees do not have to answer these question, they might do so nevertheless if ask nicely.

But the 'scandal' is still cooking, according to Focus: Flüchtlinge aus Syrien und Irak machten mehrmals Urlaub in ihrer Heimat.
But:

Baden-Württemberg seien seit 2014 rund 160 Flüchtlinge zum Urlaub in die Heimat gereist. Das stimmt nicht. Flüchtlinge dürfen unter bestimmten Umständen zwar ins Ausland reisen, aber "Heimaturlaub" für Flüchtlinge gibt es nicht.

__Translation: Since 2014, about 160 refugees in Baden-Württemberg have travelled to their homes. That's not true. Refugees may travel abroad under certain circumstances, but there is no "home leave" for refugees.
Source: BR 2017

And despite the apparent low number of either known or suspected incidents, the debate continued:

Integrationsbeauftragte rechtfertigt Heimatreisen von Asylbewerbern
Es könne durchaus Gründe geben, warum anerkannte Flüchtlinge zeitweise in ihre Herkunftsländer reisten, sagt Aydan Özoğuz. Das dürfe aber nur in Ausnahmefällen geschehen.

__Translation: Integration Commissioner justifies asylum seekers travelling home. There may well be reasons why recognised refugees travelled temporarily to their countries of origin, says Aydan Özoğuz. However, this should only be done in exceptional cases.

Source: Zeit Online, 18. August 2017

Then we have cases where people get asylee status, but decide to return home for good, for whatever reasons. They might leave the country and visit their home country, preparing their eventual return.

That constellation is still not solved in Germany, despite being constructed as of immense importance:

"Asyl-Urlaub":

Wer angibt, in seinem Heimatland bedroht zu sein und dennoch während des laufenden Asylverfahrens dort hinreist, dessen Asylantrag soll abgelehnt werden.
__Translation: Anyone who claims to be threatened in their home country and still travels there during the asylum procedure should have their asylum application rejected.

Seehofer's Master-Plan Migration, via ntv, 10. Juli 2018

And it has only recently been tackled in Austria:

Asylverschärfung: Wer im Heimatland "urlaubt", verliert das Bleiberecht 18. April 2018, 13:37 Der Ministerrat hat ein strengeres Asylrecht beschlossen. Die Regierung will damit "Missbrauch abstellen"

__Translation: Stricter asylum laws: Anyone "on holiday" in their home country loses the right to stay. The Council of Ministers has decided on a stricter right of asylum. The government wants to "put an end to abuse".

And is equally controversial still in Switzerland:

Flüchtlinge auf Heimaturlaub
Gerichtsurteile zeigen, weshalb Menschen in das Land zurückkehren, aus dem sie einst geflohen sind.

In der Schweiz dagegen strich Justizministerin Sommaruga den Passus aus dem Gesetzesentwurf, der Flüchtlingen Reisen in die Nachbarländer ihres Heimatstaates verbietet. Ob das Parlament dies in der Beratung auch so sieht, bleibt abzuwarten.

__Translation: Refugees on home leave

Judgments show why people are returning to the country from which they once fled.
In Switzerland, on the other hand, Justice Minister Sommaruga deleted the passage from the draft law prohibiting refugees from travelling to neighbouring countries in their home state. It remains to be seen whether Parliament will see this in its deliberations.

Basler Zeitung, 12.03.2018,

Pro-refugee-helper organisations in Germany now advise refugees in Germany this:

Frage und Antwort: Reisen bei subsidiärem Schutzstatus (20 Feb 2018)

In andere Staaten kannst Du ebenfalls mit dem Aufenthaltstitel und deinem Pass einreisen, wenn Du ein Visum dafür hast. Probleme bei der Wiedereinreise nach Deutschland können entstehen, wenn Du eine Reise in dein Heimatland machst und ein Vermerk im Reisepass über die erfolgte Ein- oder Ausreise eingetragen wird. Daraus kann geschlossen werden, dass dort keine persönliche Verfolgung mehr besteht und somit die Gründe für deine Flucht entfallen sind. Ein Schutzstatus in Deutschland also nicht mehr benötigt wird.

__Translation: You can also enter other countries with your residence permit and passport if you have a visa. Problems with re-entry to Germany can arise if you make a trip to your home country and a note is entered in your passport about the entry or exit. From this it can be concluded that there is no longer any personal persecution and that the reasons for your escape have been eliminated. A protection status is therefore no longer required in Germany.

There are some important and possibly confounding factors that might influence international reporting on this issue:

Germany does pay good money to refugees to return to their home country!

This is laid down in Rückkehrhilfegesetz and might correspond now to up to 3000 Euros. But this is of course not for "happy holidays" but "good luck with your life there". And apparently, getting German tax payer money to return to any home country is not that attractive:

Rückkehrprämie für abgelehnte Asylbewerber offenbar ohne Wirkung, Spiegel 27.03.2018

To get any perspective on this using simple guesstimate statistics:

Germany in 2017 had 745,155 first time applications for asylum. As official numbers are lacking, we might extrapolate from the media reported cases: over 100 cases reported from one federal state, without detailing any reasons or purposes for any of such travels, and these numbers heavily disputed in the first place.

Rough estimate for the math lets me estimate this with a dark figure of 100% as a quota of (200*16) / 745,155 => 0.43% possible cases.

Just to bind that back to the question as currently framed: not even for this small quota does the German state pay any airfare for holiday or vacation destinations, and especially not for detinations in persecuting home countries of refugees. If the German state pays for air travel, it's to incentivise the refugees to return home for good.

There are only the two possible scenarios when the German state pays for a flight for refugees back home, even regularly:

either via already mentioned return-home-premium for recognised refugees that incentivises them to return home for good or refugees that had their asylum application denied and do not get subsidiary protection status. The latter means they are flown from immigration-detention back to their presumed point of origin. At least: these are the rules. Actual outcomes for these are summarised in this FAQ: Kosten, Ablauf, Ausnahmen: FAQ zu Abschiebung (NDR, 20.01.2018)

_{Personal commentary: I do not have the slightest doubt that the blog post accurately describes the personal anecdote that the blog writer has asked and received in personal conversation the answer as reproduced in the claim. The answer he received (and consequently the claim as such) is widely believed and used in right wing circles. (Do not click the link unless you have to.)}

Conclusion:

The claim is distorting the facts that are known. There were refugees returning into those lands they claimed were persecuting them. These numbers are not officially known. The resaons for the travels are utterly unknown. Very few of the very few seem to be like real "vacations", but reasons and qualities of these travels are opaque for the government. But those cases that were reported as falling into this category of "vacation in persecuting countries" appear to be quite low and are then very widely considered "abuse of the system." But some of those trips even appear to be justified in the eyes of the government officials.

In the subquestion about airfare the claim either conflates issues or is deliberately misleading.