When one is golfing and finds the urgent need to defecate, but finds no convenient facilities, one might be forced to defecate in the cup (hole of the green). 'Hot' refers to the fact that poop is warm. 'Gopher' refers to the burrowing critters in the classic golf film Caddyshack.
"Guys, that schnitzel last night really went through me. I don't think I can make it back to the clubhouse. Mind if I hot gopher this hole?"