For most of the game (that I've experienced so far) you're required to always have Mario as your party leader. While he's a great character, sometimes I'd just much rather not have him in my party. Does this ever happen throughout the story?
It seems to depend on what game mode you are playing. If you are playing story mode alone, Mario seems to be required in the party. According to one GameFAQ, if you are playing coop, you may choose your leader.
In the story mode campaign, Mario's always on the team, as he's the team leader. Whenever I play co-op with a friend, it's the two of you working together, and you each choose your own character, so in co-op, you can use anybody and Mario's not restricted.
Another article for the game mentions that he cannot be swapped out:
Mario, who cannot be swapped out of your roster, is more of an all-rounder: he carries a blaster, a hammer, and can stomp on his enemies.
I would try out the coop mode if you have another person to play with, otherwise, it seems like Mario is always going to be there.