I think that probably varies based on the specific labels you are using, the quality of print (draft, normal, best) that you are using, and the setting of the printer. There is a place where you specify the type of paper and when I choose the labels, it seems to work fine with minimal waiting. I suppose ten minutes or so.
I have cards I've made for specific uses which are a few years old and show little wear. Then again, I didn't make them to be used as everyday tricks. Made a few for a trade show here, a special strolling thing there... they don't get used for more than a few days. With a LOT of ink (as in Tarot designs), it may require a longer wait and a bit more care in handling. If they get wet, for example, the colors will run.
It's not a replacement for "real" printing, but it's worked very well for me for one-off projects where I need a special card.