(Preface: The below answer was initially written before WOTC's acquisition of D&D Beyond on 2022-04-13. I assume the acquisition would not make my conclusion of 'treat the digital sourcebooks as official' less valid.)
I believe at least the digital versions of the sourcebooks themselves (currently listed here) should be treated as an official rules source roughly on par with the books.
Firstly, WOTC considers D&D Beyond to be an official digital toolset for the game (though, until the acquisition, D&D Beyond wasn't made by WOTC directly, but licensed through them):
This morning, Curse launched D&D Beyond—an official digital toolset for Dungeons & Dragons fifth edition.
(dnd.wizards.com, 2017-08-15)
Secondly, the books on D&D Beyond are considered to be 'digital sourcebooks', which, on their own site, are defined thusly:
A digital sourcebook is a completely digital version of one of the published books, such as the Player's Handbook or adventures like Curse of Strahd. You will get the book re-created in digital format, as well as unlock all of that book's content for use throughout the toolset - both for current tools and anything on the roadmap (such as encounter building/ combat tracking, etc.).
As they are considered a 'completely digital version of one of the published books', I'd say they're as official-source-worthy as said books.
In addition, BadEye (Adam Bradford, D&D Beyond product lead at Curse at the time of posting) also calls D&DB an official source, mentioning that their site is meant to be kept up to date with the rules as new changes come in:
Errata will be incorporated as it comes in. As an official source, it's important that we always stay current.
Ultimately, due to the above, I feel D&D Beyond's version of the sourcebooks should be considered as official as the books themselves. If there's a discrepancy between their content and the most recent errata/printed version of a book, then the book (plus relevant errata) is the 'most canonical', sure, but I don't think the potential for that kind of mismatch/data entry/out of date error would make the rules from the site overall considered 'unofficial'.
Disclaimer: The above only applies to content on D&DB from the official sourcebooks. D&DB also hosts various things that shouldn't be considered official, such as usermade homebrew, forum posts, and unofficial sources such as the Blood Hunter class from Critical Role/Matthew Mercer- these are usually kept separate in the site's navigation. The SRD & Basic Rules are also hosted there in a combined section, but even the WOTC-hosted PDF/printed versions of the SRD is considered unofficial:
The sword of sharpness deals an extra 14 slashing damage when you roll a 20 on its attack roll. The SRD incorrectly says otherwise. Note that the SRD is not an official rules source for D&D. #DnD
@JeremyECrawford, 9:54 PM - 4 Jan 2018
In addition, applicable Unearthed Arcana content was once hosted there (for the UAs released in the range of Jan 8, 2018 to the discontinuation of UA on D&DB around August 2021), which should be treated the same 'officially unofficial' way any other UA content is treated.
While the parts of D&D Beyond that are aggregates of data from the sourcebooks (such as the "Races" section, the "Monsters" section, etc.) contain reproductions of sourcebook data, they seem to be slightly more prone to data entry errors- I'd recommend using the digital sourcebooks' version in the case of finding a conflict between the two.
(On the topic of "what's considered an official sourcebook?", @nitsua60's longstanding question/answer "Where do I find the “official” rules for D&D 5e?" is worth a read as well.)
Maybe.
The AL Player's Guide and AL DMG don't address which rules to use outside of setting minimums and outlining the "+1 rule." So we turn to the AL FAQ (version 7.1 as of this writing) which contains this relevant guidance on p.2:
What Rules Do I Use?
As a D&D Adventurers League Dungeon Master, you are empowered to adjudicate the rules as presented by the official materials (PHB, DMG, MM, etc.).
So it has to be "the official rules," and PHB, DMG, MM are examples of such rules.
The next page makes a bit of an argument against, in that it lists twenty-four separate sources and doesn't list D&D Beyond, but arguments from omission are inherently weak, in my opinion.
That said, since the differences between D&D Beyond and the PHB, for instance, haven't been issued as errata you're "safe" going with the legacy wording.
Proceed with clarity.
In a few months we should be seeing a set of AL documents in the v.8 form which one hopes will make some mention of D&D Beyond. Until then just be clear with your players/GMs what you're using. Either "I don't see D&D Beyond the list so I'll stick with listed sources" or "it's an official WotC product which is what they say to use" certainly strike me as legitimate positions. As usual: what is important here is probably not so much arriving at a "correct" conclusion as is arriving at a shared conclusion.
Best Answer
There is no official order of precedence, but you should probably trust the "book" pages over the generic listings
D&D Beyond is meant to be an accurate representation of all official 5e material, and almost all of the time it is, but as with any large project there are sometimes errors that sneak in - much as the physical books sometimes have printing errors or other problems and need to be updated by errata.
Given the way the site is put together, the content which is included as a direct representation of a specific physical book is most likely the correct transcription, and errors are more likely to have crept in in the generic aggregated databases of content. As @V2Blast points out, one factor is that the statblocks in the compendium versions are basically just text formatted to look like a statblock, and so they don't have any of the technical limitations that the generic monster database does - for instance, at the time of posting, the inability to attach ad-hoc text to the skills listing to indicate conditional modifiers like the Shadow's.
Therefore, I would trust the entries as they are given in the Monster Manual, Basic Rules, or whatever other "book" representation (essentially, URLs starting with /sources/ or /compendium/) over the generic entries. However, there's no "official" order of precedence for the site's content, as these disparate pages are all meant to agree! The standalone listings are meant to match the compendium versions, except when a newer version of the same thing (e.g. a monster statblock) is published in a newer book.
If you have an account on the site, you can report issues with the site content in the support forum. The site devs seem to be pretty good about correcting errors like this when notified.