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If you are at a reputable university you can probably get legal access to nearly everything you need for research just by visiting your university's library and asking for a copy of the article. This is nearly always available to you. If you are grant funded, then grant funds can probably be used to obtain the necessary papers if the university cannot get them. In the US, even my town library has been able to get me access to things just by asking and because they have developed relationships with other (university) libraries. Small universities can have formal relationships with large research universities to "borrow" books and articles.

Likewise, borrowing the resources of colleagues is permitted. If s/he has a legal copy s/he can print it. The printed copy can be loaned to you. There are no issues with this at all.

So, the situation you describe should be rare if you do a bit of legwork.

But if you cite something, cite a legal repository, not a website known to pirate academic work. You don't need to actually own a copy of a paper to cite it, but it is probably a mistake (for your reputation) to flaunt illegal or unethical access.

It probably isn't as difficult or as costly as you imagine to do the right thing.

Would you incriminate yourself?

For some articles, the abstract is so clear and concise that it effectively says everything you need to know in order to cite it. I've previously been advised when writing abstracts for articles in paywall journals to make sure someone could cite the article without actually having it. Obviously, it's not an ideal situation, but it is very much possible to cite an article purely based on the abstract, which you would have access to without acquiring the article anyway.

In short, if I see someone cite an article and I somehow know that they have not paid for access to it, I'd assume they've cited it based on the abstract.

First, you have to cite if something is relevant for your work! It has nothing to do with how you aquired the article, and even if you do not have the article, just know the abstract or one particular result that is relevent, cite it! (big insight after I worked in academia, seldomly the articles cited are also read in entirety).

By the way, in academia authors get NO money from their articles, it is all done for reputation in the scientific community; so by not citing you actually do more harm to the individual who wrote the article then by the act of downloading (where maybe just the publisher loses money). And by the way for scientific articles the system works a little bit different, it is seldom the case that an individual buys individual articles (and if they like to they are tredemnously expensive). They are either acquired by your library through subscription, by interlibrary loan (many libraries are connected by networks), given to you by the authors themselve (once I just got a copy from an article that is hard to get in person from the authors send by post after asking him at a conference), or nowadays by the way you asked for... I will not judge what is unethical here but after reading this (or being in academia for yourself some time) you might view it a little bit different...

So, if nothing works you usually end up not getting the article at all!

Sidenote, this is a little bit different for books as the authors get some money from them, not much in academia too. But for non-academic books where the authors have to life from the money it is definitely unethical, but this is an entirely different system. Do not judge and confuse it by that (which people might do here if they compare it to robbery, netflix etc...).

Also if you do not cite something relevant some reviewer of your article will probable notice and either point you to the literature, or if it is a well-known article might conclude that you have done a bad review of the literature yourself.

Robert Columbia's framing of the question:

It seems that you are not asking about the ethics of downloading articles from questionable sites, but only the ethics afterwards - when the downloading is already a fait accompli, and the real question is whether citing said document counts as an additional unethical act above and beyond what you did when you downloaded it (e.g. "citing a document obtained through an unauthorized route").

For discussion purposes, I'm going to go contrary to the accepted answer and say yes.

First, if you're using ideas that are not your own, you definitely have to cite the source. Failure to do that is plagiarism. With respect to avoiding plagiarism, it doesn't matter where you got the ideas; credit is due.

However, inaccurately citing where you got the source makes it harder for others to trace that back to the source.

In some cases, the content posted to a pirate website looks like or maybe is labeled as a copy of the content from the original source, but in fact has different content.

I have wasted many hours trying to figure out why an author cites X for assertion Y, when in some cases they were actually citing a hidden X' which differed from X on, among other things, assertion Y. I would have much appreciated if the author had included a citation like:

J. Ixsom, P. Smiflich, and D. Seuss: "Methods for safely pipetting oxyflogated dexlahydrates at STP." 23rd Annual Conference on Unusual Chemistry (CUC '17) London. Retrieved Jan. 18, 2019 from https://authorsite.org/pdf/1234.56789.

Then I have all the usual citation information about who published that and where, and in most cases I can go to that source and get it from them. In this case, the publisher gets an extra sale from the author having cited that work, regardless of how the author obtained it. However, if I can't find support for what's being asserted, I can then trace through to what the author actually looked at. If there's a difference, I can more quickly debug. I might see how the author was led astray and be able to resolve the issue another way. Without that link to what the author is actually citing, though, it's very hard to determine the basis for that assertion, and requiring that extra work is a negative impact caused by a difference between the source the author claimed and the source the author actually used.

Also, note that content differences are not all malicious, and the "authoritative" record is not always better from a content perspective. Sometimes, authors posting a file on their own website, with the publisher's permission, will update those files (e.g. correcting a mistake in a formula) and then the formula being used in the citing article looks different than the one it's being cited for. (To authors: If you do this, please explicitly call it out as a correction.)

Sometimes, the differences are honeypots intentionally inserted by publishers designed to get the message out that only official sites can be relied upon for accurate information.

Just cite the paper properly (e.g. take into account where the article was originally published), obviously don't cite Alexandra Elbakyan. Seriously, nobody cares and nobody can find out anyway. And the people who care have a very bad case of boy scout/teacher's pet disease. The editorial houses are rent-seeking rackets that profit from publicly funded research and hard labour of academics. And they still have the nerve to paywall it. Anyone that feels bad about avoiding that needs to do a deep reflection on their moral priorities.