This one is underrated. In LA, at least, it also became kind of rivalry among Hispanics involving your "keep it real" types on one side and your "go to college, get a good job" types on the other. Kind of like a B-Hop, Donovan McNabb thing. Saw it in a bar in LA and the atmosphere was great.
:blood Pac/JMM- Donno about heated but I think this deserves a little mention. Frazier/Ali Ward/Gatti Zale/Graz? Floyd/Pac (Even though they will never fight, their crap has been going on for years) Calzaghe/Hopkins :hey
The most popular ones were already mentioned. Even when most boxing fans are not happy to confess, but Haye-Klitschko is probably one of the modern rivalries....
DLH - Vargas was bitter. but rivalries are more edgy when both guys are on the same level talent wise, thats why MAB - Morales is so good.
Lewis-Rahman and Lewis-Bowe got very heated and personal (even with the latter not resulting in a professional fight) Louis-Schmeling went way beyond "heated" due to the political climate of the time, but the fighters themselves became good friends