Will it hurt to exercise with a fever of 101?

Probably: This partly depends on the cause of the fever; it it is a virus then your heart may well be affected & exercise could create serious heart problems like cardiomyopathy. If it is bacterial the risk is less but your body needs energy to fight the infection & exercise places added stress on our bodies. 101 is not really high; if you feel up to it you could go for a walk, but avoid strenuous exercise.

Take it easy: The fever is body's way of telling you that it's responding to something. To be honest unless you're a professional athlete who has to "perform" because it's your job, listen to your body and rest a bit. That's doesn't necessarily mean stay completely bed bound but it would be wise to give your body some time to convalesce.