I don't think apostrophes will become extinct. It's true that many people do not know how to use them properly, but that doesn't mean they aren't necessary. If people didn't stop at a stop sign it wouldn't make the stop sign unnecessary, it just makes people wrong. The apostrophe has a specific and necessary use in the English language. Since that use (the need to show possession) will not disappear and the need isn't fulfilled by any other grammatical convention., the apostrophe isn't likely to become obsolete.
I would think that speech marks are more in danger than apostrophes. We already have texts where speech is denoted by hyphens and the conventions seem to be flexible across the world. Though the apostrophe is misused, it is necessary for understanding and I think will be retained.
I find that there are many things that change about English. It is a living language, and I am always amazed at the artful way writers (even song writers) use the language.
We have seen the more prevalent use of "there's" as with "there's" lots to do today, which makes me crazy—but though around for some time, it's not the rule as much as the exception. The misuse or absence of apostrophes shows a complete lack of concern for using the language properly. I don't think they will go away, but I do believe that we will be seeing more and more poor writers who learn by texting.
Some people are not going to use the apostrophe the way that it should be used. There will always be those kinds of people. Then there are people who cringe (as do it) to see these words misspelled (it's/its is the worst for me). It's about caring enough to learn, to remember, to know! There will always be those kinds of people, too. These problems with apostrophes may continue, but if those hiring are the ones who took the time to learn how to use them correctly in the first place, the "newly educated" may find themselves missing out on jobs to those who DO know how to use them. (We can only hope.) I like "the little guys" (apostrophes).
Laziness and a fundamental lack of understanding is no reason to eradicate the usefulness of the apostrophe. I fully recognize that one of the most often misused grammatical devices is the apostrophe and the inability of students (and, apparently, many adults) to differentiate its uses in possessives and contractions. But its elimination will require some other way of telling the difference between words like "it's" and "its". Americans certainly won't stop using contractions, and people'd sure miss out on identifyin' the colloquial speech of good ol' Southern slang if th' 'postrophe ain't 'round t' use.
I would hope that they do not. They are used for specific reasons. That said, I am rather sick of correcting apostrophe errors. It seems that the errors made are normally made out of laziness (at least with the papers I grade). But, it does not seem any different than the failure to capitalize or use text lingo in formal papers (grrrrrrrrr).
The grandparent of English, French, in which Chaucer wrote much poetry, follows the same form as the German. So, since Old French and Old English were cousins, the possessive case was indicated in the same manner, by putting the possession first and then the owner. e.g. The book of Geoffrey. [as opposed to Geoffrey's book]. Incidentally, this is where the term o'clock comes from. It is ten o'clock/ of the clock=10:00.
Rather than regressing to the Old English, Americans in their indolence and unconcern for standards may soon dispose of some of the symbols whose meanings have become confused. Indeed, it is interesting that George Orwell wrote that "Language does not reflect culture; language is culture." If he is correct, English is in danger.
My opinion is that it probably won't disappear. Its appearance was required by the change in English when the genitive case endings were dropped that were used in Chaucer's time--or--in the time of Chaucer. I see only two options if the apostrophe were to be dropped. One is that the possessive must revert to the Germanic form that influenced English in early times, as English is a Germanic language. Instead of Pete's dog, we will have to have "the dog of Pete." Another option is the return of genitive suffixes such as Chaucer used in writing about the Knightes tale, where the apostrophe eventually came to mark the ellipsis of the /e/ in -es in the genitive/possessive Knightes leaving Knight's. I don't foresee that English speakers/writers, infected as all are with idolization of progressivism, will ever revert to any past thing, thus making it necessary to keep the apostrophe. I do foresee that society will have to a much better job of teaching the apostrophe, which first requires a much better job of understanding the apostrophe.
I wish they would so that I would no longer have to see people misusing them. I am heartily sick of seeing apostrophes (or should I say "apostrophe's") used to make words (word's) plural. YUCK! So, I guess I would not be sad to see them go...