MPU6050 safe to use with glue?
Would this have any possible conductivity issues 


Nope.


  or risk damaging the wires or board itself?


Nope.


  If it is OK to use the hot glue, would it be possible to cover the entire MPU6050 in the glue without compromising performance. 


Sure.

This is known as "potting". You can get special potting compounds, but hot glue is just as good.