Patrons have other sources of information, but we can also ask if they can just see through feign death
I completely agree with PJRZ’s answer, but it is a bit of a frame-challenge so I wanted to attempt to provide more of a straight answer. That is, we know a patron most likely won’t be “fooled” because various external factors (e.g. a book, a talisman, possibly a bond forged by the pact itself) give away the lie, but the question can still be asked, “Does the patron rely on these things? Or are they also able to see through the spell itself?”
No explicit statements about this kind of thing
This is a difficult question to answer because warlock patrons are defined rather vaguely. Archfey, celestial, fathomless, fiend, genie, great old one, hexblade, undead, and undying, are each fairly broad categories, and many of those categories are very vaguely defined in and of themselves. For instance, the great old ones and fathomless are largely defined by being unknown. And those that are better-defined are rather broad—the lowest fiends are definitely not capable of seeing through a magic disguise; they’re not even capable of independent thought (but then, those are near-certainly not capable of being a warlock patron, either).
Evidence that being a patron does not automatically mean you can see through feign death, even of your own warlock
But I would offer a suggestion that there is some evidence that patrons do not, as a general class, have the ability to automatically pierce mortal illusions, transformations, and the like. This wouldn’t normally matter as they are certainly very powerful, and likely to easily make any saving throw a mortal forces them to make, but feign death does not allow for a saving throw. So in order to see through feign death, they need some kind of ability that overcomes transformations, like truesight.¹ Some patrons have something that does that—and I don’t mean, say, archfeys can but fiends cannot. I mean some archfeys can, and some archfeys cannot; some fiends can, and some fiends cannot. Same for all of the others (though, again, great old ones and fathomless are so little-defined it’s hard to be sure there).
The closest we get to an explicit statement is to look at the stats of the creatures that are described as potential patrons.² For instance, the fiend patron lists balors and pit fiends, which both have truesight. On the other hand, Xanathar’s Guide to Everything includes a unicorn as a possible celestial patron, and that doesn’t have any special senses that could plausibly see through feign death. Most patrons, however, are not statted, and even those that are, are suggested to be more powerful than the normal stats for a creature of their kind would suggest (fiend patrons might be “pit fiends and balors that are especially mighty,” genie patrons are “noble genies [who] wield power that rivals that of a lesser deity,” etc.). The unicorn isn’t described in such a manner, but it’s possible to see that as implied by the very suggestion it could serve as a patron.
For that matter, though the fiend patron is described in the Player’s Handbook as some of the most powerful possible fiends, if a unicorn can do it, I would expect that a succubus can, and again, no truesight for them. Succubi are considerably more powerful than unicorns, and are very much geared for this exact kind of thing. Certainly, 20 years ago, a succubus was the example “fiend of corruption” depicted in the official art,³ which might be the last time that D&D has actually given mechanical rules to the fiend’s side of the process of granting a mortal power in order to corrupt them.
So I don’t really think it’s necessarily the case that all patrons could see through feign death, that they are all powerful enough that truesight or similar can be assumed.
My basis for this is that, generally speaking, warlock patrons are something not quite as powerful as a god, at least in terms of ability to give out magic power. There can be some overlap—the most powerful celestials, fiends, and genies can “rival lesser deities” as Tasha’s Cauldron of Everything puts it—but even though the high-end patrons can be comparable to low-end gods, there are still lesser patrons in each category who certainly aren’t.
Generally speaking, gods produce clerics, and warlocks are largely the province of those for whom clerics aren’t an option. Some of the most powerful patrons may have both—at least one being³ is both a fiend and a god, for that matter—but generally speaking, it seems that clerics are easier for a god to produce, and do a better job of feeding power into a god, than warlocks are for patrons. Clerics are largely required to put in the effort for their own training, usually provided by other clerics, and they are both strong sources of faith and prayer, and also tend to bring others to the faith and thus generate yet more prayer. Gods are powered by faith. Warlocks, by contrast, expect “something for nothing,” and the patron has to personally reach out and empower them, and for the most part patrons are not gods and so are not inherently rewarded for their warlock’s pact—the warlock has to go out and do something to be worthwhile to their patron.
And I bring that up because if patrons are generally sub-god-tier, if only the most powerful patrons in each category reach those heights, then they probably don’t, as a class, have an ability that gods lack. And it does seem that gods lack inherent truesight. The case here is murky—gods haven’t been statted in D&D in nearly 20 years.⁵ But at least then, clearsight, which was a “salient divine ability” that gave permanent truesight, was a specific ability that some gods had, not an automatic part of the package. There are also numerous occasions within D&D history wherein a mortal has fooled a god, even their own god.⁶
Jeremy Crawford states that feign death cannot be seen through via truesight, pointing out that feign death is not an illusion. However, feign death does magically transform a creature into an apparent corpse, and a creature with truesight “perceives the original form of […] a creature that is transformed by magic.” I think Jeremy focused too much on the illusion aspect of things and forgot truesight can do more than that.
The only “this individual actually is a patron” that we have stats for is Zybilna, an archfey. She has truesight out to 60 feet.
Fiend of corruption was a “prestige class,” a class that could only be taken by multiclassing after meeting certain requirements. In this case, fiend of corruption required one be a fiend and have charm person or charm monster abilities—at the time, there were both playable fiends and monsters with class levels. Anyway, it was printed in Fiend Folio, April 2003, a year and a half before Complete Arcane was published with the original incarnation of the modern D&D warlock. So of course fiend of corruption didn’t use the terms “warlock” or “patron,” or provide abilities that were identical to those a warlock got. Still, extremely similar idea. Ironically, after Complete Arcane, the warlock became one of the best ways to qualify to become a fiend of corruption—but then, that seems fitting to me.
Lolth is a demon prince as well as a goddess—she came to that position because she was an elven goddess (then known as Araushnee), was kicked out of the pantheon and lost her divinity (for her betrayal of Corellon Larethian), became a demon prince, and then recovered her divinity. In most cases, celestials and fiends are mutually exclusive with divinity, though the most powerful of these blur the lines, and several books—especially recent ones—don’t make firm distinctions. Asmodeus is often called a god, but at least in my opinion, this is an abbreviated statement that doesn’t get into the full details of what Asmodeus is—most likely, he does not have (and does not want) a proper divine portfolio. He has deity-tier power, but without the strings attached that divinity would entail: why would he choose to bind himself?
Deities & Demigods, April 2002. While it’s a convenient source for the idea that not all deities have truesight, it shouldn’t really be relied upon for anything—statting the gods was a mistake then, and they haven’t done it since for good reason. Wizards basically never considered it truly canon—deities in D&D have always been exactly as powerful as the adventure called for, no matter how powerful other books have claimed they are, and discrepancies were just waived off as one or the other (or, more often, both) being an avatar or similar. If I knew such an example off-hand, I would far prefer to cite some novel or something where a deity was fooled by a mortal illusion or transformation, instead, because Deities & Demigods is not a great book.
It didn’t involve illusion or transformation, but Andromalius, the Repentant Rogue, is a vestige from Tome of Magic whose entire story revolves around the grand trick he pulled on his own deity, Olidammara.