Recipes say not to boil butter for beurre monté - why?
Cooking doesn't equal boiling, mind. For example, sous-vide cooking cooks something at very low temperature.

Beurre monté is a sauce of emulsified butter, which will remain emulsified up to 80-85ºC (more or less). If the temperature goes up, your sauce will separate into fat (~80%) and water (a little less than 20%), will lose texture and you also might end up with rancid flavors if the temperature goes up too much and the fatty acids in butter hydrolize (as in break down with the help of water)