Forrest Gump doesn't see it but Lt. Dan has a lot of anger towards him and generally treats him with contempt. But underneath it is an appreciation that Forrest holds no grudges and will do anything to help a person in need. By the end there is no doubt they are best friends.
In Guardians of the Galaxy, this Trope could describe the relationship between pretty much any two members of the team (except Groot, who's friendly to everyone), but especially between to Drax and Gamora. In one scene, the former calls the latter a whore and then a few minutes later shoots Nebula with a rocket launcher when she is talking trash to Gamora and planning to attack her.
Kiss Kiss Bang Bang has Harry and Perry. Harry is sure that Perry is really his friend, and somehow ends up being right. Harry would give as good as he gets, except he's a colossal ditz and no-one can out-insult Gay Perry anyway.
The Boys in the Band. Michael and Harold. While it isn't clear they are best friends, two moments reveal their deep relationship: Michael's gift to Harold, and Harold's quiet assurance after giving him a "The Reason You Suck" Speech "I'll call you tomorrow."
Will Turner and Jack Sparrow. The principled blacksmith and scheming pirate are frequently at loggerheads, but in the first movie Will risks everything to rescue Jack from the gallows, and in the third, Jack sacrifices immortality to save Will's life.
Jack and Barbossa begin to show shades of this in On Stranger Tides.
In Made, the two main characters are best friends from high school, but they bicker almost nonstop due to Bobby's short temper and conservative approach in contrast to Ricky's loud, reckless and unreliable personality.
In My Girl, Vada and Thomas J are best friends and throughout the movie, Vada can be seen fighting with him and other times she is seen having fun with him. All in all, Vada cares deeply for Thomas J.
Luke's relationship to Han Solo. Especially when they're arguing over Leia.
R2-D2 and C-3PO qualify as well.
In The Lord of the Rings trilogy, Legolas and Gimli are portrayed in this manner. Basically Legolas is Bugs Bunny and Gimli is Daffy Duck. As much as Gimli was exasperated by Legolas and as often as Legolas would act smarmy towards Gimli, the two always ended up being there for each other. This is an exaggeration of a similar relationship in the books.
Gimli: "Never thought I'd die fighting side by side with an Elf." Legolas: "What about side by side with a friend?" Gimli: "Aye. I could do that."
Sam French and Jake Wyer in 50/50 have this in spades, although Jake likes to pretend he can't stand Sam.
Dante and Randall from Clerks. They both argue about things all the time and snark at each other (especially Randall, who seems to enjoy messing with Dante), but when it's all said and done, the two are almost inseparable buddies. Jay and silent Bob have a similar dynamic, except Jay does most of the talking and Silent Bob ignores it with an air of sarcasm, despite not saying anything.
Jorge the conquistador Sanchez member and Carmelo the native Mesoamerican Sanchez member from The Book Of Life, have a rather complicated relationship with each other. The former often calling the latter savage and the latter unintentionally bumps around the former. Not surprising considering the complicated history between the conquistadors and Mesoamericans. They're still family though.
In Teenage Mutant Ninja Turtles (2014), Raphael is constantly at odds with Leonardo, hates being told what to do, and mentions that he's leaving the group the first chance he gets. However at the end, he defers to Leo's leadership and tells the others to do the same, as well as confessing that he says he's leaving because he's really just scared that he's not as good as them, and he only says that to make them bond harder when it feels like things are driving them apart.