Certified GKH Gardening Expert
It's best to cut out the dead branches and see if the tree recovers on its own.
If you suspect the tree might be diseased, though, you might want to remove it to prevent any disease from spreading to nearby related trees (other stone fruit trees). Make sure to sterilize your tools with a 10% bleach solution after you prune the tree to kill any potential disease organisms. Or, consider talking to an arborist or local extension agent to see if the tree has any disease problems.
Also, see these articles about plum tree diseases: