Can gum infection cause fatigue symptoms?

Yes. Any infection can cause one to feel bad, sluggish or ill. It would depend perhaps on the severity of the infection. Oral infections are usually painful and can become systemic in the body. One has to be careful that they do not have certain cardiac or valvular conditons of the heart that oral infections could make very much worse. If concerned see your dentist or oral surgeon and your pcp.
Possibly. There are increasing evidence that periodontal disease can cause or contribute to many problems and symptoms, including fatigue.