First of all, extreme cold might mean anything depending on your personal taste. What I mean by extreme is any temperature under minus 30 celsius.
Secondly, I have no idea, whatsoever how to play any instrument in that hard condition, but tried it a couple of times and here i'm trying to pretend that I'm the smart guy, who know how to do it, but actually I ain't go no clue.
Rule Nr. 1: Don' play the Jew's harp under -30 C. Don't be an idiot! Nobody will be listening, it's not fun, it's not romantic, it won't be musical at all. Stay at home, drink a chai, watch the frozen streets from your window, listen to the wolves, but don't put steel in your mouth.
Rule Nr. 2: In case you really have to play Jew's harp in that harsh condition, let's say because you are facing the barrel of a Kalashnikov, or your really dear host had super kindly asked you and thinks that it's cool, or you have an untamable pervert desire to try extremities, then think of rule Nr. 1.
Rule Nr. 3: Do it with dry lips, otherwise the instrument freezes to your lips.
Rule Nr. 4: Do it quickly. I don't mean quick playing, but don't waste your time on finding the right key, say your prayers to the instrument, then turning three times clockwise and facing to the Sun. Just take off your glove (one only), grab the steel, pluck some notes and this is it.
Rule Nr. 5: Don't use massive breathing techniques. It is not healthy to inhale minus 30-40 Celsius air right into your lung. You can easily get cold. So, breath in through your nose, exhale through the mouth.
Rule Nr. 6: Don't try virtuoso hand techniques. Your sensation in your fingers will be gone after 10 seconds, you won't feel if you cut yourself, actually it's hardly possible to use your fingers properly for more then a minute in such weather conditions.
Rule Nr. 7: If you are from Yakutia, don't pay attention on my words. These hints are for those Western heated living room/studio/stage jew's harp player pussycats like myself. The nordic wolf khomus tough guys will lough at this.