Assuming your garage door is the 'Up & Over' type, i don't think cutting the handle off would be much help really. Try giving it a good hard push at the very top of the door, in the middle (directly above the handle). That's where the door latch is, and it clips into a housing screwed to the frame above. The screws tend to slacken a little over time, allowing a little movement of the housing which may be enough to let you pop the door open without any damage. Its worked well for me 3 times in the past. If you do get in this way, when handle is fixed remember to tighten the screws in housing - stop thieves getting in the same way!
DougT, October 2009