Can brick patio and walk edging be removed without damaging the edging?
You could remove it, but your patio may start to shift eventually if not replaced. The paver edging  is there to keep the pavers in place. There are many different types of paver edging, and bringing it up might pull up your lawn a bit and possibly your pavers. 

Instead I would try to hammer the edging back in by placing a board over the edging and tapping it back in with a rubber mallet. If it is thin edging I might try to cut off the dislodged sections with utility knife.