Episodes suggesting a conflict
In Season 1, Firey slapped Coiny 246 times. In Season 2, Firey and Coiny had not yet slapped each other.
Firey and Coiny are constantly fighting, and in most episodes when they are seen interacting, they usually fight. Their way of fighting is slapping each other (seen above).
In Take the Plunge: Part 1, while standing on Spongy, Firey angrily replies to Coiny that he is dumb. In response, he slaps back and replies that he is easy to slap, then slaps multiple times, before being stopped by Golf Ball.
In Don't Lose Your Marbles, Firey says that he is so use to slapping Coiny, that he still has to do it.
In A Leg Up in the Race, Coiny was eliminated. Since that episode, Coiny has made a few cameos: In The Glistening, Coiny votes for Firey to be eliminated, and Firey tries to slap him, but fails. In Hurtful!, Metal Firey was slapped by Invisible Coiny. In Return of the Hang Glider, Coiny votes for Leafy, and said he would be upset if Firey won Dream Island. When Firey won, Coiny fainted.
Episodes suggesting a friendship
In Return of the Hang Glider, Coiny fainted when Firey had won. However, Firey let him onto Dream Island when he threatened him to let him on and explained why. And when Blocky comments on how "that was the first time Firey and Coiny agreed without fighting", they hug because they both agree that Flower should stop from destroying all recovery centers. However, in Season 2, they seem to be enemies again (see above). Despite this, the tensions between the two seem to be calmed to an extent. They don't get into slap fights at all anymore, and while they express anger for each other, neither of the two goes to major extremes to do so.
In BFDI Is Back, Firey and Coiny seemed to become good friends with each other, since they never had an argument with each other since the beginning and end of the episode. Firey even apologized to Coiny while slapping him and Coiny on the other hand, never was hostile to Firey anymore. But this is mostly non-canon, meaning they probably will get in a fight again when BFDI is resumed.