Generally not, but rarely if it dumps sufficient snow (isnt it like a 11 to 1 ratio of volume between snow and rain??) it potentially could or form enough on the ground over time to flood the fire. (fast drifting snow is another issue)
Generally shelter is first thing you look for, even out in the open when there is no convenient cave/overhang (that under the pine tree idea or rather up near the trunk as a windblock) Wind Chill will suck heat out of you faster than a fire will normally provide (then I will build a bigger fire .. and spend time in the wind chill getting the fuel for it ....). SO cold outside (40 below) a fire can only do so much for you.