This is a question that has been handed down for generations. Do you toss it? Recycle? Make a fancy pot holder out of them?
Well if you have a 5 month old puppy following your every move, this question is easily answered. Leave it on the floor and let her have into it.
I changed out my old, beat up grips last night and put the old ones on the coffee table and my dopey dog came over and took them both off the table and over to her bed. she had a grand ol' time chewing on one, i think its loosening up another one of her baby teeth.

So, if you have a dog that likes to eat anything, donate your old grips to a better cause. They will appreciate it, and the corners of your furnature will as well.