Should I remove unused screws from my frame?
If you're a weight weenie then your frame wouldn't even have those mounts.

Removing the screws might save you 10-20 grammes, which is less than a good dollop of mud, or under half a snackbar, or a quarter the weight of your spare tube.

If you're worried about rust, remove screw and apply some grease then refit. 

If you're really worried about rust on the bike then fit mudguards, for which you'll need those screws anyway.

Answer: don't bother removing spare screws/bolts from your bike.