It doesn't directly, his department is known to be a money pit. But having him at their hospital is kinda a point of prestige for the hospital; kinda like reputation padding. In a way, you can look at the budget for his department as a marketing expense for the hospital.

They get to claim they have a world famous diagnostician on staff who can figure out what's wrong with the most hopeless causes.

It's a teaching hospital, so they don't (entirely) rely on the regular patient-funded system common in the USA.

House's reputation can totally help with funding.

It'd be kinda like having a really prestigious professor at your university.