How Much Tire Will Fit Under My 71 Dart

As 70aarcuda said, it all depends on what you can/want to do. If you don't want to go to the work and expense of mini-tubbing, you can maximize the tire size by using offset spring hangers and shackles. This will only gain you about .75-inch per side. You can get a 275 tire in if you optimize the wheel backspacing.
Now if you want to go to the work and/or expense of tubbing, you can get any tire you want to fit under there. Years ago I raced a '69 Dart that had 14x32 slicks completely under the car. Full tubs and the outer wheel opening was not raised but was stretched 4" to get the slicks to fit.