In general the answer is, no, $100 bills do not get more valuable, because we don't really need them in the first place.
However, it is technically possible for that sort of thing to happen. For example, let's say Americans needed quarters to do their laundry, and there was (for some reason) a major shortage of quarters. If it was the only way to get your laundry done, you might consider trading away a dollar for the three quarters you needed, in which case we could claim the value of a quarter had risen from 25 cents to 33 cents.
If people burned most of the large denomination banknotes in a society where cash was the only form of money, and no new notes were issued, that might create deflationary pressure. Someone trying to sell a car for $10,000 could find no-one could scrape together enough cash to make the transaction, so give up and sell it at a lower price. When prices go down, money is effectively more valuable.
But we live in a society where there are many convenient alternative to $100 bills (bank transfers, $50 bills, etc) so the lack of them wouldn't make much difference.