Does the resistance of a coil increase if it gets burnt?
It could go either way.

If the cross section of the coil decreases, due to chemical combination of the surface metal into non-conducting oxides, then the resistance will increase. This will be permanent.

If adjacent turns of the coil touch, then the resistance will decrease. This may be permanent, but could be reversed if the coil moves again and they stop touching.

An LCD to monitor 'techy' stuff about the vape sounds a bit excessive, but I guess you've got to distinguish it from the competition somehow.