You can't damage synchros riding a clutch. You'll wear out the clutch disc, throw out bearing, pressure plate, and burn the flywheel surface. Synchros make sure that each gear is spinning at the same speed as you shift, whether it be up shifts or down shifts.

Shifting without the clutch is hard on synchros because you aren't disengaging the engine's power from the driveline, thus causing each synchro to work harder.