In answer to your question yes they can.
On most drives the circuit control board and I/O interface of the drive sit on the underside of the drive it`s self.
The bare circuit points touching any form of metal will short circuit the control board Matthew.
If for example you have ever removed a 2.5" laptop hard disk drive you will see a light weight shroud made of tin or aluminium, if you unscrew this from the 2.5" drive you will always see a thin plastic sheet between the drive and the base of the tin or aluminium shroud.
And should use or put some thing non conductive between the base of the mount and the circuit board of the drive.
Clear tape, a thin plastic layer, card paper, acetate sheet or any other thing that is plastic and thin.
I`m glad you had the wisdom to think about it being a problem Matthew.
You obviously pay good attention to what you are doing.
Just out of intrest, FYI. yes I have seen people short circuit drives by doing the exact same thing as putting the drive circuit controller board directly down on to a bare metal surface.