No other source book currently printed overrides the core rules (unless you want it to)
Other than the PHB, DMG, and MM, the only other source book that resembles a general rule book (as opposed to a book describing a setting, adventure module, or other additional content) is Xanathar's Guide to Everything (XGtE). And XGtE explicitly says in its introduction that it does not override the core rules:
The options here build on the official rules contained within the Player’s Handbook, the Monster Manual, and the Dungeon Master’s Guide. Think of this source as the companion to those volumes. It builds on their foundation, exploring pathways first laid in those publications. Nothing herein is required for a D&D campaign — this is not a fourth core rulebook — but we hope it will provide you new ways to enjoy the game.
In short, all of the rules in XGtE are optional rules. Taking an example from your question, the falling rules given in the PHB are quite simple:
At the end of a fall, a creature takes 1d6 bludgeoning damage for every 10 feet it fell, to a maximum of 20d6. The creature lands prone, unless it avoids taking damage from the fall.
That's it. The rules don't even say when you fall, they just say what happens when you do fall, and mostly leave it up to the DM to determine when falling happens. XGtE gives some additional (optional) mechanics to use for falling, none of which contradict or override the PHB's rule quoted above. In fact XGtE begins its section on falling by restating the PHB's rule itself:
The rule given in the Player’s Handbook is simple: at the end of a fall, you take 1d6 bludgeoning damage for every 10 feet you fell, to a maximum of 20d6. You also land prone, unless you somehow avoid taking damage from the fall. Here are two optional rules that expand on that simple rule.
Of course, even in cases where XGtE contradicts the core rules (one well-known example being magic item crafting), there is no question of precedence, because XGtE explicitly cedes precedence to the core rules in all cases by declaring that all of its rules are optional. For example, it's entirely up to the DM whether to use the DMG's crafting rules, XGtE's crafting rules, or something else entirely.
To summarize, there is no general rule for which books take precedence over which other books. Instead of defining a general rule that is external to the rule books, the books themselves tell you which ones take precedence (with the only current example being XGtE).
Non-book rule sources
Of course, there are other rule sources besides printed books, as well as sources of things that are not rules. Here is my understanding of how they fit in. (Thanks to the many commenters for bringing most of these up.)
Errata
Errata issued by WotC are corrections to unintentional errors, ommissions, and the like in a specific source, and as such always override the source they are correcting. They don't change the order of precedence between sources, so for example an erratum for XGtE would not override a rule in the PHB.
Wizards of the Coast provides a free "Basic Rules" PDF that contains selected content from the core rulebooks (mainly the PHB and DMG, I believe). Unlike XGtE, which is explicitly positioned as a a companion to the core rules, the Basic Rules document is designed to stand alone, such that you could (in theory) play D&D 5e using only this document, albeit with a relatively limited set of options. As such, the Basic Rules do not declare any explicit relationship to the core rule books. Generally there should be no problems arising from this ambiguous precedence, since all the rules in the document are excerpts from the core rule books, so in theory there should be no conflicts, except where errata issued since the publication of the Basic Rules (dated November 2018) have caused the latest printings of the core rules to diverge. Obviously, in such cases, the errata and/or corrected printings of the core rules would prevail.
D&D Beyond
All printed sources are also available in online form on D&D Beyond. While D&D Beyond is run by a 3rd party, they have licensed the official materials from WotC. The content on D&D Beyond should always correspond to the latest printing of the corresponding book, with all official errata incorporated into the text. In other words, there are no new official rules presented on D&D Beyond; it's just an online source for the same rules that are already published in printed form.
It's worth noting that in addition to simply reproducing the content of each source book, D&D Beyond also collects content from multiple source books into single pages for specific topics. For example, the D&D Beyond page on the wizard class includes all the relevant content from the PHB as well as all the additional arcane traditions published in other sources (at time of writing, this includes the War Magic tradition from XGtE and the Bladesinging tradition from the Sword Coast Adventurer's Guide). Every piece of information on the page is reproduced from a source book, but there is no single source book that contains all of the information on this page. For the most part, this shouldn't matter, but if you do run into any issues relating to rules precedence on these multi-source pages, you should resolve the issue using the original sources.
This is an official publication from WotC, but it contains official rulings, not additional rules. The goal is to clarify how the existing rules are intended to work, not change them. As with XGtE, this is explained within the Compendium itself:
Official rulings on how to interpret rules are made here in the Sage Advice Compendium [...] A Dungeon Master adjudicates the game and determines whether to use an official ruling in play. The DM always has the final say on rules questions.
Jeremy Crawford's tweets (and other public statements by WotC)
The Sage Advice Compendium further explains that it is the only source of official WotC rulings, and no ruling is considered official until it appears in the Compendium:
The public statements of the D&D team, or anyone else at Wizards of the Coast, are not official rulings; they are advice. Jeremy Crawford’s tweets are often a preview of rulings that will appear here.
(Jeremy Crawford's tweets in reply to rules questions were previously considered official rulings, but that has changed recently. See here for information about the change.)
Unearthed Arcana
WotC occasionally publishes play-test documents referred to as Unearthed Arcana. As with other WotC-provided materials, these documents always clearly explain their relationship with the core rules. For example, the 2019 revision of the Artificer class says:
This character class is presented for playtesting and to spark your imagination. These game mechanics are in draft form, usable in your campaign but not refined by full game development. This class is not officially part of the game.
In general, official rules will never be published in Unearthed Arcana. Any UA that does become official does so by being published in an official source book, at which point the official publication should take precedence over the UA version.
Sage Advice D&D website
There is a website called "Sage Advice D&D" that is unaffiliated with WotC. It has nothing to do with the official Sage Advice Compendium. Among other things, this website collects rulings made in tweets by the D&D designers. However, as noted above, these tweets are neither rules nor official rulings.
(Note: I have no experience with Adventurers League, so some of the above may not be correct for AL.)
