I agree that asking for a sentimental item of little or no monetary value is not rude, if done at the proper time/place, since it is more about remembering the person. Asking for valuable or large items, such as sets of furniture, is more about trying to benefit yourself.
POD to this - and I think Calistoga had it right in her earlier post, as well. If you have a valid sentimental reason to request something, and you can find a polite way to do it, I think it's fine to put it out there that "If no one else is attached to that shadowbox of old keys that Grandpa had hanging on the wall behind his rocking chair, I'd really love to have that for my living room" - but that assumes that the shadowbox has minimal value and it's not something everyone would have shared memories for.
My grandparents moved to a retirement home together a few years ago, which involved significantly downsizing their belongings. I was the only grandchild who was unable to come divvy things up. (I'm the oldest and was the only one out of college and married at the time.) I really only requested two or three things, and I'm thankful that my mother was there because one of my cousins wanted everything I wanted (and then some). I should have realized that all of us grandkids would have shared memories of the same toys in Grandma and Grandpa's basement! My mother pointed out that my cousin was already getting X, Y, and Z, and I wasn't there to claim anything other than the few things I had asked for ahead of time.
My cousin is still a bit put out that I got the old rocking horse (like this one
but painted yellow and much cuter - named "The Coy Colt"
) - it was my mother's when she was a little girl, and dad and Grandpa touched it up for Babybartfast. Mind you, at the time, my cousin was single, still in college, and had no storage space, while I was pregnant with Babybartfast
I'm definitely going to pass it on to a family member when we're done with it, but it will probably be to one of my siblings!
ETA: found it!
That's Babybartfast from quite a while ago (before she had any hair), but isn't the rocking horse cute?