Absolute minimum you need is a tower that has a CPU cooler clearance of 165mm. And thats assuming you have RAM 32mm or shorter. If you dont, you'll need to add the height of the RAM (over 32mm) to the overall height of the cooler, since you will need to adjust one of the fans up to account for the added height of the RAM. This is because the cooler overhands the RAM slot and the fan will interfere with the RAM.
Example:
Say you have a set of Ripjaw V RAM. This ram is 42mm tall. Since the fan will need to be adjusted up 10mm to compensate, you would need a case with a minimum of 175mm CPU cooler height clearance. That's 165mm for the cooler, plus 10mm from the ram to equal 175mm total height.
If you're willing to spend that much money on the cooler, you might want to go for a case that isn't super expensive. Chances are you wont find many cheap cases with 165-180mm clearance.