They bite when they play, but out of plain disrespect, no way they do. The only time I see my horse bite our other horse (who is alpha) is when he is playing. Horses establish themselves in the herd every single day.
If you always have treats in a certain pocket or if you have them in your hands then yes. They can get mouthy. I have seen it plenty of times. It is a fact taht horse can start to bite when they are given a huge amount of treats. There is nothing wrong with treats but it is something that can happen.