No, the A/C compressor has its own ribbed belt. So, yes you can leave the belt off. But, you can certainly install the belt and leave the electrical connector disconnected. I think if you simple turn off the compressor (with the snow flake button) there might be instances where the climate control system will activate compressor anyway (i.e humidity control). But, the system has a low pressure cut-out switch so it probably would not engage anyway.

Be warned that the A/C compressor tensioner pulley adjusting bolt is a real PITA. Many of us cut a notch in the end of the bolt so a screw driver can be used to snug it up (or loosen) before final tightening. once you look at it, you will see what I mean. This might only be an issue if you have Dynamic Steering, Sport, etc.

If the compressor clutch does not engage, then there is no harm in having the belt installed. The clutch bearing is sealed/self lubricating. if you have the compressor rebuilt, ask the shop to fill with the proper viscosity oil. If you purchase new, you need to do your research and fill with proper amount and the proper viscosity PAG oil. The proper oil viscosity (i.e. 43, 56, 100, 150, etc) is CRITICAL. If you find data for total system oil quantities, then assume 1 oz of oil remains in each major component (other than the compressor).

Good luck.