What could possibly happen if I overmix the butter and sugar in a cookie recipe?
Your butter can melt through friction heat. 

You said that you are used to melting the butter. There are cookie recipes which require this, usually chewy cookies. Creaming the butter with the sugar produces a different texture, and you should not change the method between recipes. If your recipe already asks you to cream, you should never cream so much that the butter starts to melt (which can happen at rather low temperatures), as you won't get the texture of fluffy creamed cookies. The problem only occurs with real melting, softened butter is OK. In my experience, it is actually best to leave the butter out at least overnight before creaming, instead of only leaving it out for about an hour to get soft to touch, or cutting cold butter to whippable pieces. 

While melting butter is the "upper limit" of creaming, it is unlikely to run into this problem unless you forget a stand mixer turned on, or you softened your butter at high room temperature (above 30 Celsius). With 22-ish degree butter and a hand mixer on moderate speed, your arm will go numb before your butter melts.