mjhjmtu

I've come across these two sentences and have been trying to wrap my head around them for a while.

「出られたくないんだよ」and 「縛られたくないんだよ」

Both use the potential verb form and 「くない」
When I look up the translation, they say "I do not want to leave" and "I don't want to be tied up." But they end with 「られた」, which is used for something done in the past. I would have assumed they'd be translated as "I did not want to leave." If I'm missing something please tell me.

I think you are confused. That has nothing to do with past form. It is called ～たい form and it indicates "being willing/wanting to" do something.

The basic rule of construction is:

verb stem of masu-form + たい.

In your example, the verbs are conjugated in a passive negative form (as for 出る notice that passive and potential are the same so I'm unsure just from the verb without context).

Basically you can see it as conjugated doing the following steps:

縛る　ー＞ (passive)　縛られる　ー＞　(〜たい form) 縛られたい　ー＞　(negative) 縛られたくない

For more about the ～たい form, you can look here.

Ps.

I got caught up in something and could not continue my answer.

I wanted to add one thing. I did not mention translations because you have already provided them. However, I'm not sure "I do not want to leave" is correct. Then what would be the difference with "出たくない"? Since the 出る is intransitive, this is probably what is called 迷惑めいわくの受け身, loosely translated as "troubling passive". Therefore, is probably more something like "I don't want (someone else) to leave".

However, put in this way, the more I think of it the more I find it confusing myself. Let me expand. Think of this type of passive as something that happens and someone being affected/troubled by it.

Typical example:

雨に降られた, I was fallen on by the rain (It rained on me).

And I was affected negatively by this event.

Now let's turn this example to make it more similar to your case:

雨に降られたくない, I don't want to be fallen on by the rain (I don't want it
to rain on me).

Now should be easier to understand why I said the translation of your case should probably refer to someone else leaving. There are probably two things missing there:

[私が][...に]出られたくないんだよ, (I) don't want (...) to "leave on me"

Where I wouldn't know how to better render in English the "leave on me" part, but I hope it gives you the idea.

And obviously, now you should see why in English is much more natural "I don't want (someone) to leave".

After seeing Naruto's comment, I think "sufferer passive" is an even better translation for 迷惑の受け身.