Searching for the reason that the gear lever on your Mercury outboard motor won't shift the gear case is a process of elimination. Like any troubleshooting and repair effort, finding the cause of the problem begins with the simplest, least expensive solution and works its way toward the worst case scenario. This means that solving the problem may be as simple as replacing a shift cable or as complex as completely rebuilding the gear case.
Disconnect the remote control cable from the engine and try to move the shift handle with an adjustable wrench. If the shift handle is hard to move, the problem is either in the shift box or the cable that leads to the motor. Disconnect the cable from the control box and take it to your marine supplier to secure a replacement cable of the same length as the one you removed.
Connect the new control cable to the control box -- but not the motor -- and operate the shift lever. If shifting is still hard, the problem is in the control box: remove the screws that hold it in place and replace it. If the shifting feels normal with the remote control cable disconnected from the motor, the problem is with the lower unit.
Wrap the jaws of a bench vice with soft cloth. Remove the bolts located around the flange at the top of the lower unit that hold the lower unit to the motor, using a socket wrench. Pull the lower unit away from the rest of the motor by pulling straight down, to avoid damaging the shift lever or the driveshaft. Set the unit in the bench vice and tighten the jaws enough to hold the lower unit upright and stable.
Inspect the propeller and its rubber hub visually. If the hub is shredded, loose or slipping, or appears otherwise damaged, replace it with a new hub. Replacement of the hub requires a trip to a propeller shop equipped with the proper tools and experience for such work. If the hub and propeller are undamaged, verify that the problem is inside the gear case.
Have an assistant turn the propeller clockwise as you move the shift shaft on the lower unit back and forth between "Forward" and "Reverse." If the gears do not engage, visually inspect the shift shaft for damage; replace the shaft if necessary. If the gears engage, the problem is not in the shifting mechanism at all, but in the gear case itself, possibly a broken or damaged gear. This requires disassembly of the gear case and replacement of the damaged parts.