I had a Japanese Plum tree that I cut down and removed the stump but I am still getting "volunteers" coming up across my lawn. Is there a way to get rid of these without destroying my lawn?

Are the 'volunteers' suckering off the old roots? If so, your tree isn't as dead as you think. If you don't want to destroy your lawn, your only option is to cut the suckers down to about two inches and carefully paint the little stumps with full strength brush killer. You will be at it for awhile.

Are they comming up from seed? That's a lot easier to deal with - just pull them up, making sure to get at least the top of the root. If you leave any little stems, they will resprout.

Whether they are root suckers or new sprouts, mowing should be all that is needed. Eventually they'll run out of energy to keep trying.

I agree with Purple. I find anything in the prunus family sends out root suckers and I just mow them (in the lawn) or snip them at the base (in the garden beds and around the trunk of the tree). I also have a healthy stand of quaking aspen that is determined to take over the world, but they are also easy enough to either mow or snip. The key is to be diligent.