Well - I've done it, and I'm in a climate where we never need heat - but the engine is definitely not happy like that.
In my case it was a beater that I had just had a sheetmetal thermostat cover fabbed up for, put it together with the stat and it leaked, so I tossed the stat - I only needed the car for another 90 days (had one in the bodyshop after a smackup), it was burning oil, leaking coolant, and just drinking fuel - but you know what - it got me from A to B - and I was sorry to see it go.