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I requested permission for MATLAB's trademarked logo usage on my site, https://dragonnotes.org/ , from MathWorks (trademark holder) - who responded,

"Due to the high number of these types of requests that we receive, we are unable to evaluate and respond to each request on an individual basis and thus unable to provide permission."

What are the implications for my site? Does MathWorks still hold the right to take action, and what would such an 'action' constitute?

I am assuming that you are in the United States for this question. Please correct me if this is not the case.

MathWorks still holds the right to take action, which may be anything from cease-and-desist letters up to litigation.

You have followed the correct process in asking for permission to use a trademark. The owner of the trademark, MathWorks, has given you their answer, which is quite simply, "no". You may not feel that their reasoning is fair, but the default state of trademarks is that they are under ownership of whomever created them, and you do not have permission to use them. Things will remain that way unless MathWorks changes their mind.

It doesn't matter if it feels dismissive of them; they are under no obligation to even consider requests to use their marks. If they didn't explicitly say, "sure, go ahead", or even, "yes, you may use it provided you follow a list of conditions", then using their trademark will be an unlicensed usage.

If you still would like to pursue getting permission to use their trademarked assets, you will need to try to contact them again. Until they say otherwise, using their trademarks will be considered unauthorized use.

(Edited to add this clarification brought up by @David Siegel):

However, your usage might not be violation of trademark.

The primary purpose of trademark restriction is to stop someone from misrepresenting a product as being from the entity that owns the trademark; this stops someone from, for example, selling a cola soft drink called "Coke-a-Cola". The reasons for this are manyfold, but the basic idea is that allowing that type of usage means that consumers might not be able to tell that your product is distinct from the original, and could then mistakenly attribute the quality and level of service of the previous brand with the new product.

If your usage of MATLAB marks is such that you are identifying the products used as from MathWorks, and not yourself, and are doing everything in good faith to disassociate your website and/or offerings from MathWorks, it is possible that your usage would be considered correct usage of trademark.

Even if it is legal usage, MathWorks still may decide to take action.

If MathWorks believes that your usage is unauthorized and that it is trademark violation, they may decide to take action. This is regardless of whether it actually is; until you have this case in front of a court, you will not get a definitive answer.

We cannot answer whether this is a legal usage of trademark.

Ultimately, whether or not a usage of a mark is considered to be correct usage is a question that can only be answered by the courts, which means the only person who can give you concrete advice on a course of action is a lawyer.

In lieu of proper legal advice, you will need to weigh the risk of MathWorks taking action against your usage with the benefit you receive from usage.

(Edited to add this clarification by @Dale M):

Regardless, you may be breaking copyright by using the logo.

There is a separate issue besides just trademark at play here. The copyright for the MATLAB logo belongs to whomever created it/owns it (presumably, MathWorks in this case). Using the logo without permission is a copyright violation.

The only case in which this would not be a violation is if the logo is released for use in general under a compatible license, such as Creative Commons; do note that these licenses typically have additional conditions, such as requiring attribution.

If you are unaware as to whether there is a such a license, or if you fail to follow the terms of the license, usage of the logo almost certainly constitutes copyright infringement.

That response means that you do not have MathWorks's permission to use their trademarked logo or logos. If your usage is such that you in fact need such permission, MathWorks could indeed send you a cease-and-desist letter, or file suit against you, and might win such a suit.

However, not all uses of a trademark require permission.

Suppose XCorp makes a razor, and I make replacement blades that fit that razor. I might use XCorp's trademarked logo for the razor to show what brand it is that my blades fit. In that case I would need to make it very clear that I was not sponsored, authorized, or approved by XCorp, so that no reasonable consumer would be confused into thinking that I might be.

Or suppose I want to review and compare several different makes of razor. I might use the trademarked logos of XCorp, YCorp, and ZCorp to identify the makes that I am discussing. I still need to make it clear that I am not endorsed or approved by any of these firms, but it will probably be easier to do that in this case. That is particularly true if the reviewer is not selling anything.

Both of these are examples of Nominative Use where a trademark is used as the name of the trademarked product, service, or firm. A non-owner may use a trademark to identify things that his or her product or service is compatible with, or that s/he is discussing in some way, provided that no reasonable person would be confused or misled into thinking that the non-owner is in any way authorized, endorsed, approved by or related to the trademark owner.

I can't tell from the question plus a very quick look at the linked site which logos are from MathWorks, nor exactly how you are using them. If you are not selling anything, or you make it very clear that the logos are used for identification, but that you are Not MathWorks nor approved by them your use is quite likely perfectly legal, and would have been had you never asked for permission.

However, to be safe, you would be wise to consult a lawyer with trademark expertise, to make sure that your use qualifies under this concept.