Can I replace the washer by just shutting off the hot water?
YES- on 3-handled tub/shower valve units the supply water is delivered seperately to the hot and cold sides; turning off the hot water supply will allow you to work on the hot side valve.
Because multiple units will be affected you should do some research on how these are repaired prior to turning the supply off, to minimize down time.
Your description of the problem ("leak from the hot handle... when the water is running") is indicative of a problem with the valve's packing. I recommend rebuilding the whole valve while you have the water turned off (bibb washer, packing, grease the stem thread, seat if necessay, etc.) Keep in mind that there are many (many, many, many) brands, models, and configurations out there and there are no "standard" parts. Washers come in a bunch of different sizes and types.