As with the typical Facebook "grammar" post, these complaints constitute a very mixed bag. Many, if not most, are not really (from a linguist's perspective) about "grammar" at all, but rather about ancillary issues: spelling, punctuation, and so on. For example, the difference between it's (the contraction of it is) and its (the possessive determiner) is strictly a spelling issue. There is no doubt that English speakers know the difference, that is they have separate lexical entries for these homophones in their mental dictionaries. And there's no doubt that when English speakers say it's a shame they know they are producing a contraction of it is a shame; when they say its handle broke they are referring to something's handle, maybe a cup: the cup's handle broke.
Other sets of homophones can be dealt with similarly: there their they're; too two; four for fore; and so on.
One of the "crimes" Weird Al mentions: the confusion of, who and whom. This is a bit different. The distinction here reflects a time when English was a more highly inflected language, with word endings signaling the case of a noun or pronoun. The English noun boat had forms that included bāt for singular subject and object; bātas for plural subject and object; bātes and bāta for singular and plural possessives respectively; and bāte and bātum for singular and plural use with prepositions.
Like 'boat,' once upon a time the English interrogative/relative pronoun who (earlier hwā) had different forms for different cases. Three of these have survived: who (the subject form); whom (the object form), and whose (the possessive form).
And, as with the forms of boat, a leveling process has been going on over the centuries. We now have only three forms for boat: boat, boats, and boat's. The last two are now homophones, so phonologically there's just /bot/ and /bots/. Somehow, though, English speakers know when they mean 'more than one boat' versus 'belonging to a boat'.
As for the forms of who, there's been a leveling process going on here as well. Who saw you? still works, of course, but Whom did you see? is now almost always Who did you see? This is one of those things that makes the grammar nazis cry, but it's perfectly normal. Languages change. And English has changed quite a bit, as anyone who tries to read this sentence will realize:
Fæder ure ϸu ϸe ært on heofonum, si ϸin nama gehalgod.Insisting that people keep using whom even though, in normal English, it has nearly disappeared, is not unlike trying to make people go back to using fæder for father. Kind of silly.
In a book titled On Literacy, Pattison (page 162)* noted that trying to force people to use forms of language that have fallen or are falling out of usage constitutes "nostalgia as repression." This is an apt term for this form of what amounts to abuse on the part of people who profess to be language experts, but who in fact are sadly undereducated in that area.
*Pattison, Robert. 1984. On Literacy: The Politics of the Word from Homer to the Age of Rock. Oxford University Press.