I don't know about McDonald's and Burger King. Maybe a few decades ago, but there are so many other hamburger chains out there now that the rivalry between those two doesn't have the same impact.
Coke v. Pepsi (which sounds like a lawsuit!) is the biggest corporate rivalry I can think of...they consistently try to top each other or copy each other with each new innovation or advertisement. Vanilla Coke..Vanilla Pepsi...Caffeine Free Coke...Caffeine Free Pepsi...Pepsi Clear...did Coke do clear?
IBM v Apple was another good one, although I suppose it is "PC v Mac" now.