Depends on the rim width - try not to go narrower than the outer rim width and you should be ok. Unless you inflate your tyres over ~100psi then you may also suffer from tyre "flex/roll" if you have a narrow rim and a high volume tyre.

Also any 700c tyre is the same diameter as a 29" mtb tyre and a 622etrto tyre. Gives a few more options.