Let's get one thing straight. Ground-Fault Circuit Interrupter protection for personnel is designed and intended to protect a human (or any animal I guess), from being electrocuted (killed) due to an electrical fault. In most situations, the grounding system will handle any direct faults to ground. A GFCI devices is there to protect you, if you happen to come into contact with an energized conductor.
Let's also be clear that a GFCI device may not protect you from a shock, but should protect you from death (electrocution).
Examples:
You grab hold of the ungrounded "hot" conductor.
In this situation, the GFCI device should disconnect the circuit before you are killed.
The frame of a device is energized, and for some reason the grounding system has not performed its job, and you come into contact with the energized frame of the device.
In this situation, the GFCI device should disconnect the circuit before you are killed.
A GFCI device is there to protect you, from death.
But why does it seem like codes tend to require GFCI protection in locations where there is water, like bathrooms and kitchens?
The simple answer is that when your skin is wet, it's a better conductor. Because of this, you could potentially provide a fairly good path to ground. It's also important to provide this protection in bathrooms and kitchens, because it's common for plumbing in homes to be grounded.
Let's say you're washing your face in the sink. The water is running, as you reach for a towel. You reach with one hand to turn the water off, while the other fumbles for the towel. While you're still holding the faucet with one hand, the other hand accidentally finds the non-GFCI protected receptacle. The electricity races through your wet hand, across your body, and out your other hand into the well grounded faucet. While crossing your body, the electricity ran right through your heart causing it to stop. Now, you're dead.
If it was a GFCI protected receptacle, you would have gotten a bit of a shock and a scare. But you'd still be alive.
As for the question asked...
Put an imaginary box around the tub/shower, and extend it all the way up to the ceiling. If any part of the exhaust fan is within the box, you should GFCI protect the fan as per the installation instructions.
If it makes you feel safer, then by all means GFCI protect the fan.
Best Answer
The casing is more than likely nailed or screwed to the ceiling joist on the side that does not move. It may be possible to open a small slot along the face of the joist. Then run a reciprocating saw (Sawzall) along the face of the joist, eventually cutting the attachment fasteners and freeing the casing.
The risk here is you could damage other things attached to the joist, such as wiring. If that happens, you'll need to do the alternative approach. Partly to access the fasteners and partly to repair the damage.
The alternative approach is cut out a big enough panel in the ceiling to give you room to work and remove the fasteners anchoring the casing. Patching the hole isn't too much work, though matching the texture finish can be tricky. You may have to settle for a less than perfect match.
If grounding is at all possible, such as bonding to a nearby metal cold water pipe, it should be done. However you are not required to retrofit a grounding system to your house to replace one fan. If there is no reasonable grounding means, leave it unattached.