We see conflict all over the place, all the time. Teammates fight. Parents fight. Coworkers fight. Partners fight. People who are on the same team fight each other. It happens. Folks with the same goals and motivations bicker, scrap, punch and kick each other for lots of different reasons. But no matter what the reason, such conflict always makes for interesting stories.
There have been plenty of great duos and teams in movies. And their partnerships are all the more interesting when they have some volatility sprinkled in. The characters Eddie Murphy and Nick Nolte played in the 48 Hours movies couldn't be a worse fit if they were a square block trying to go into a round hole. But they had common goals and could stomach each other enough to not kill themselves and get the bad guys.
When penning a screenplay where your heroes and heroines get along as well as Tom and Jerry do, be sure to establish important aspects of their character and their motivations. Audiences like seeing protagonists fight while striving to get what they want.
Write your protagonists as having different personalities. In Captain America: The Winter Soldier, Captain Steve Rodgers is an almost antiquated but endearing patriot while his fellow protagonist Nick Fury, however, is gruff in his belief that the freedom America enjoys has to be ensured with bigger guns than the rest of the world. In order to have conflict like that be meaningful between your characters and to your audience, establish their juxtaposing personalities first.
Give them spotlight scenes early in the screenplay before (or as) they fight that demonstrate the kind of characters they are. We see Captain America playfully run laps around a healthy veteran and Director Fury shoot and swear his way out of an ambush in addition to seeing the two protagonists argue over how to save the world. We know who they are and how they do things so that when we see them butt heads, we know why and how severe it is.
If you're writing a screenplay where the heroes aren’t famous comic book legends, it'll be harder to establish who they are in regards to each other. So if you're creating original characters, establish their relationship to each other after you establish their individual personalities. The Misters in Reservoir Dogs are a good example of several protagonists being introduced and having conflict over how to attain a shared goal. Why are they together chasing the same goal? Answer that question with dialogue between them. Tarantino accomplishes an interesting intro in the first scene by having his colorful characters talk about a range of different things at a diner before he has them introduced to their diamond heist details and aliases.
After you establish your protagonists, their relationships and their goals, pick a conflict and go with it. In the Rush Hour movies, Detectives Lee and Carter always have a B story-like conflict going on between them. Either they don't trust each other or, as seen in Rush Hour 3, they aren't as close as they thought they were. Make sure to pick a conflict and write your characters saying and doing things that address the conflict, make it better, or exasperate it into a bigger problem.
No partners ever have an entirely smooth time chasing their goals. There are external and internal conflicts that they do battle with in order to the save the day, even if your characters aren't superheroes, detectives, or jewelry store robbers. And the more conflicts your protagonists have to navigate, especially between each other, the more compelling your screenplay will be.