Many of us are in the same position. I think ultimately we will all go the same path.
First see if a library, school, collector--anybody--wants them. If not, keep the few that you feel personal connection to, and, yes, trash the rest.
I used to carry mine around with every apartment move. Then I decided to trash all of them. After my last move, about 7 years ago, I started collecting them again. Truthfully I never look at them, and they sit in their Playbill binders, taking up so many shelves. Ultimately, I know I will trash them all again rather than move them. (with my kindle, I've already been getting rid of my book collection.) I have four playbills with autographs (one signature in eyebrow pencil!). I'll keep those.