It is when someone is being especially stubborn. Derived from the saying "You're as stubborn as a mule." and from the Irish being so stubborn. I'm Irish, I would know.
Dave: There is no way I will ever watch that TV show.
Scot: Why not? It's a good show.
Dave: No, I refuse to.
Scot: Come on man, don't be an Irish Mule.