Smartphone mounted on handlebar: Will vibrations during cycling damage it?
The vibrations will be hard on it.  The result will not be instantaneous failure,  but an increasing likelihood of failure after perhaps several hundred hours of riding.

The likelihood of damage can be greatly reduced with a resilient, shock-absorbing mount of some sort (I assume most commercial mounts include some shock-absorbing function).

Most important is to avoid mounting in a way that the device will bang against the handlebar, or rattle in its mount.  A mount that is too flexible can actually increase the G force the phone is subjected to.