Normally No. Chewing gum as a repetitive action can actually increase symptoms. Most people who chew gum chew on only one side of their mouth. This actually is like exercising one arm vs the other so the muscles on the right, for example, become stronger than the oral muscles on the left. An imbalance in occlusion is exactly what you don't need!
You should not. Most TMJ symptoms are muscle related. So, if muscles are fatigued, they need to rest. Gum is a form of repetitive use injury. It should be avoided, especially when the system is already showing signs of distress.
May make TMJ worse. Tmj problems refers to problems with the jaw joint in front of the ear (temporomandibular joint). Typically the problems arise from some form of mechanical breakdown in the joint due to various jaw/teeth issues. If one constantly chews gum and the underlying condition is not treated, this will lead to more problems down the road and worsening of the current symptoms. Your jaw won't get better.