Burt and Eddie had a deal where Burt was entitled to his share of the winnings, and that deal didn't end just because Eddie played with money that was in his own pocket. Eddie refused to pay because he wasn't going to let Burt profit from him since he was the one responsible for Sarah's death.
That's why Burt told him he'd better not walk into a big time pool hall again after "welching" at Ames. Which brings up another question concerning the "sequel" to
The Hustler
,
The Color of Money
. It was never explained if Burt was dead or that he and Eddie had mended fences over the years, because Eddie did walk into a big time pool hall. That made me wonder what became of Burt.

I imagine that he never played pool for big money ever again.
You need to see the sequel then, "The Color of Money".

Burt owned Eddie. Eddie walked out owing Burt money. Burt is part of the shadowy underworld where if you owe, the juice runs., So Burt is probably figuring interest, or juice, on the debt.
Eddie is not the sharpest tool in the shed. He plays on pure emotion, every decision he makes is on an emotional basis. The final confrontation with Fats is for Eddie to do these things:

1. Prove to himself he can outlast Fats
2. Revenge on Burt by showing him that Eddie didn't really need Burt at all
3. To work out his rage at Sarah's death.
   He didn't care what happened to himself afterwards, hence the threat to kill Burt if Burt didnt kill Eddie first. Eddie could have come in with a couple friends to back him up, but he doesn't think that way. I imagine that Eddie could not beat Fats again, after the end of the movie. Eddie gave it all on that match. I imagine that he never played pool for big money ever again.

To Bert (Burt?), he still owned Eddie, like he had owned Fats. Obviously, Eddie felt differently.
Watch Fats expressions after he concedes the 2nd match to Eddie. His first defeat in 15 years. Fats doesn't seem very depressed at having lost to Eddie, who this time around is far more respectful. But Fats seems very deflated at the admittance of being owned by Bert.

Even Eddie was confused. He asked something like, "How do you figure that?" Bert's just letting pride get the better of him.

Yeah. One might think the business relationship had ended but Burt thought different. Sometimes business relationships on the street don't follow regular contract law.