Elbow soreness that is caused from boxing training could be a potentially serious issue caused by ligament or tendon damage. The elbow is a dynamic joint with a lot of muscles and bones and wires and stuff, and is subsequently prone to trouble. When a problem occurs, rest is essential until the soreness goes away. If the pain doesn't alleviate after about two weeks, or if you suspect a serious injury, you should visit your doctor. Upon return to training, care should be taken to avoid re-injury.