Can I remove the drywall in the area below the door to my attached garage?
If that's an exterior service door, you can do as you like with the drywall. Cut it out at a convenient location and replace it with BC plywood or other suitable material, or simply overlay a board. I've never seen horizontal drywall in that position before, and consider it a bad decision by the builder.



This original answer was given with little context in the question. I mistakenly assumed that this was an entry door connecting to the residence. 

The drywall is probably there as a required fireblock. You should overlay plywood or other suitable material and leave the gypsum in place. Feel free to replace the drywall first if it's in bad condition, though technically the joints must be taped with at least a rough coat.