It depends on the type of scholarship and how the scholarship gives you the money.
The school makes out a budget for you to cover tuition and living expenses and basically you can receive financial aid up to this set budget amount. So even if you have a full ride tuition scholarship, you can still receive scholarship money for living expenses as long as it is within your set budget. If you receive scholarships that exceed your budget, then yes the school would apply that money to your tuition (so the school would pay less). Basically they will not (cannot by law) give you more than your allotted budget.
The way around this is if the scholarship gives you money directly. This would be considered more like a "stipend" and your school can't touch it. The same goes for private loans, if the bank makes the check to you then you're gold. If they make it out to the school then it will just be applied to your budget, subtracted from your financial aid. Hope it makes sense, this is how it works at my (state) school, but you can check with your FA office to see their policies.