I have claimed many games that were in giveaways and haven't ended up playing them. Is there any way to remove them from the library?

No. Click the dropdown arrow, choose Hide Game, and forget they're there.

You can open a support ticket to ask for the games to be removed from your library, but typically support agents aren't amenable to such requests.

Agreed, hiding them is better than pestering them to permanently remove games. And there's a myriad of reasons you don't want your games easily visible. It doesn't cost you anything. For me, i found point&click, some games more annoying or some games just too old to really bother with.

Regardless, unless you go out of your way to find the games they won't show up.