TSH does not directly cause symptoms. It would not be the cause for having a low grade fever. However, if your low TSH is associated with having low levels of the biologically active thyroid hormones, Free T4 and Free T3, then there could be an association. Hypothyroidism is a known cause for having recurrent infections, which of course can cause a fever.
There could well be other causes, but it would be good to rule in or out the possibility of hypothyroidism. If you will please tell us about any other symptoms you have and also post your thyroid related test results and reference ranges shown on the lab report, we will be glad to help interpret.