Why does the landing gear control have 3 positions (up/down/off) on Boeing 737-800?
With the gear handle in the UP position, the retraction side of the landing gear actuators remain pressurized. The gear will stay in the full up position, however they will not be hanging on the mechanical locks designed to hold them in the up position. They've simply hit the full up mechanical stop. 

When the handle is placed in the OFF position, hydraulic pressure is removed from the retract side of the gear actuators, allowing the gear to settle down and onto the mechanical uplocks, which are essentially hooks that just hold the gear up. 

There are a couple reasons why it's beneficial to select OFF. One is that if they remained pressurized, then the gear are essentially hanging in space, at a mechanical stop provided by the actuator. Hydraulic actuators are delicate components, and these weren't designed to hold landing gear for an extended time in the retracted position. Imagine the loads imposed on the actuator in turbulence for example, as the gear are loaded vertically with each bump. Removing hydraulic pressure allows the gear to hang on the mechanical uplocks that were specifically designed to hold them, without putting additional wear on the actuators.

The other reason is that gear actuators are major users of hydraulic fluid volume, and the actuators and the lines that supply them are a potential for leakage. If they were to remain pressurized for a several hours long flight, that increases the amount of pressurized lines and components that can leak and cause a complete loss of the hydraulic system.

Note that if you look at more recent Boeing airplanes such as the 777, 787, and 747-8, you will not see an OFF position on the gear handle, only DN and UP. The OFF function is done automatically on these models. After you select UP and the gear are all fully retracted, there's a short time delay and the system automatically depressurizes the retract side of the actuators. It removes one After Takeoff checklist item for the pilots to do.

To answer some of the questions posed in the comments:
If you selected OFF while the gear were still in transit in some intermediate position between up and down, they would simply freefall into the down position since you've removed the retraction power.
If there's a leak in the landing gear actuator or its associated hydraulic lines, it would cause a loss of that hydraulic system, and all other users of that system would be failed. The 737 has two (and a half) hydraulic systems. The half is a standby hydraulic system that's very limited in its function, but I digress.