So many to choose from. Here are a few off the top of my mind.
DC Heroes in Marvel and their rivals:
Superman- Wrecking Crew, Apocalyse, Rhyno, Absorbing Man, Juggernaut, Red Skull
Batman- Norman Osborn, Doc Ock, Mysterio, Shocker, Purple Man, Kraven, Dr.Doom, Taskmaster
Wonder Woman- Loki, Enchantress,
The Flash- Hmm hard to say. Electro, Hydro Man, Arcade, Modok with his villains, Venom
Green Arrow- Bullseye, Daken, Kingpin, Sabretooth,
Green Lanterns- Thanos, Terrax, Skrulls, and all those cosmic or space villains obviously.
Booster Gold- Kang the Conquer
Marvel heroes in Dc and their villains:
Thor: Hades, Ares, Black Adam. Darkseid
Hulk- Doomsday, Solomon Grundy, Lobo
Daredevil & Punisher- They would get most of Batman's villains.
Wolverine- Deathstroke, Zealot, Vandal Savage
Captain America- Lex Luthor, Ra's Al Ghul,
Black Widow- Amanda Waller, Lady Shiva, Chesire,
Spiderman- Captain Cold, Weather Wizard, Killer Croc, Catwoman,
Namor- Black Manta,
Silver Surfer, Guardians of the Galaxy, Nova Corps - All of the Green Lantern's villains. Sinestro, Red Lanterns, Parallax,
Ironman- Amazo, Doctor Ivo, Toyman, and others.
Dr.Strange- All the magic villains like Faust, Eclipso, and more.
Fantastic Four- Trigon, Darkseid, Brainac
Ghost Rider- These two will clash over what souls are innocent and what are now.