Many safes have mechanisms that prevent force to the handle from damaging the lock within the safe. Sometimes this is as simple as a roll pin (or cotter pin), and sometimes it's a complex clutch mechanism.
It really depends on the design of the safe whether or not drilling would be required. I had an older fort knox with a forced handle, and I was able to coax the door open by leaning the safe over on its side. Turns out that it was only a simple roll pin that needed replacing. The problem was that it took almost 3 hours of labor to fix, as Fort Knox had welded everything instead of using bolts.
Drilling a safe is usually not something that will damage it beyond use, so long as it is done professionally by somebody who knows what they're doing. We drill safes on a regular basis, repair them, and put them immediately into service again.