Due to his position as a B&R central figure, he isn't just a customer. The fact he's using the restaurant means he actually let a prostitution ring growing inside his school. Quinn and his girls couldn't do all of this without his approval. Moreover, he has to cover his own butt. As a married man of wealth and reputation, he can't let a scandal erupt without taking measures to protect his integrity. Quinn isn't a crime lord with guns, high level relationships and henchmen. She's a student whom apparently has grown in a toxic environment. Her authority and nuisance potential is very limited (just like Lily's lovers pointed out). Burke, on the other hand has wealth, influence, authority and relationship. If shit goes down, he won't just stay passive, letting Hots have a catfight.
That was the whole point of Quinn and Mona's discussion. A man in his position can't let the girls have her ways without taking personal assurance. If the real tuition is a real thing, it's quite unlikely the restaurant is lucrative enough to secure tuition for several students. Burke taking a sponsorship role makes sense.
Quinn & Riona may actually have a far more serious problem. Riona's angst might be justified, if Quinn's aggressor is revealed to be more than a regular junkie but part of a rival group. Riona being afraid that Vinnie talked to someone isn't trivial considering the guy is revealed to be a bigger drug seller than Quinn.