Farting does expel some amount of body heat.

Im sure long-distance runners might relieve themselves without stopping, as it cools them down while also a release and embarrassment which might spur their brain into flight response.

I doubt if the heat loss is significant but when actively working out or doing intense work the mind might get tricked into thinking that it benefitted.

Anecdotally when running a fever it does feel better emptying out. So not sure if physiological or psychological effect.