One thing that can happen is it burns out. With inadequate supply the circuit
may malfunction and pass full supply through the servo's motor, until it overheats
and gets damaged.
If the device has a datasheet or specifications, read those. If unsure either
avoid the unknown behaviour, or be prepared for failure. Small servos aren't
going to break the bank to replace of course...
Also note that if one servo works out-of-spec, another identical one might not -
there is "manufacturing spread" in performance and the specifications are
the worst case performance.