There are plenty of objects in a hotel room that (hopefully) get a lot of attention from the cleaning staff. You can pretty much rely on clean sheets and toilets, but small items like tv remotes may not get as much love from soap or bleach as they should.
But while small objects might not be kept too clean, it doesn’t mean they need to be touched. Instead, use a plastic bag from the ice bucket to protect you from any germs that might still be living it up on the hotel remote control. And since those bags are see-through, you can still flip channels to your hearts’ content.