The Thing is generally portrayed as this when fighting Hulk, and the same can be said of Captain Marvel and Superman.
Namor the Submariner and the Hulk were both mainly used as obstacles for heroes in early comics, which may be why writers decided to pit them against each other. Hulk hates Namor's rudeness and arrogance and Namor hates Hulk's childish behavior and stupidity. The rivalry remained even when they were on the same team.
Admittedly thanks to their time on the Defenders they act more like squabbling siblings as opposed to actually trying to kill each other.
Wolverine is also seen as Hulk's rival. That's probably because Wolverine debuted in Hulk's book.
Speaking of Spider-Man, a more dangerous example would be Spidey and Venom. Even after Venom has "reformed" they're still fierce rivals.
One of the hallmarks of Superman/Batman comic book series is that the titular characters are ideological rivals and the reader is treated to their differing opinions of the same event. For example, where Superman sees the discovery of Kara as a love story (finding a long lost family member) Batman believes finding another Kryptonian is a detective story (is she really the noble person she is claiming to be, or is she hiding something?). While both have very different opinions, they generally respect each others beliefs due to their superficially different, but surprisinglysimilar origins.
Ashley Swift is The Rival to Jennifer Mays in The Maze Agency. Jen's success is all the more galling for Ashley as Jen used to work for her.
Book 3 of Les Légendaires introduce a group of hero named the Fabulous (consisting in asian-like swordman Michi-Gan, Comanshawas warrior Shaki and Michi-Gan's foster daughter Toopie) who started out as this to the Legendaries. Notably, while the Fabulous displayed the typical jerkass rival attitude toward the Legendaries at the beginning, it was arguably justified: not only were the Legendaries at this point still blamed by everyone for the Jovenia Incident, but the Fabulous also reproached the Legendaries to have disbanded for two years in reaction to the blame, thus not being there when Alysia needed them (Danael later admits they were right on this point). On the other hand, the Fabulous placing themselves as replacement to the Legendaries while calling them outdated and constantly criticizing them didn't really made them easy to sympathise with at first. After the two teams were forced to collaborate, they eventually developped mutual respect, and are in good relationship in the following books.
Jason Todd sees entire Batman Family as his Rivals. Most his plots are primarily based off of his desire to prove he's better. He constantly says that his aim to be a better Batman than Bruce Wayne ever was.
As Robin, he was rather competitive with his predecessor, Dick Grayson. Though it was more of a one-sided rivarly, since Dick feels/felt friendlier towards Jason.
Damian Wayne outright declares Tim Drake is his rival for Batman's approval. Tim (while less hammy about it) feels the same way towards Damian.
Back to Jason, he outright declares to Dick (who is now Batman) that he is the competition.
The New 52 seems to make it an Informed Attribute seeing as Jason acts less hostile and competitive to the Family. Though in a flashback (with him as Robin) he does say to Nightwing that life is a competition.