Not that I own any screen-used props, but I can kind of relate to this topic. I collect Silver Age comics and old toys. I've met some legendary comic writers and artists, but I would never ask any of them to sign my old books! For artists, it's much better if they have prints that you can buy and have signed, or if you're lucky, they'll do a quick sketch for you if you provide paper of backing board. For writers, I'll try to find an archive or reprinted comics for them to sign. If all else fails, there's the souvenir program you get at most conventions.
As far as props go, it would probably depend on how iconic the original prop is. There is no way I'd let anyone write on a screen-used lightsaber or Indiana Jones hat. For more common items, I guess it would all depend, but in general, I'm leaning toward No. I like the idea of displaying original props with a signed ticket stub, polaroid, or other momento of the event.
Even some of the stuff I'm currently making, I'd have trouble asking someone to write on. I'd love to meet June Foray, but I don't think I would ask her to autograph the Talky Tina I made, unless maybe it was on the bottom of her shoe.
Another thing you have to consider is what kind of pen you're using and what the item is made of. Someone mentioned that sharpie markers fade over time. If you use them on certain kinds of vinyl, the ink will spread out and blur over time.