You can think of your car's heater core like a radiator in reverse. A vehicle's heater core carries hot engine coolant and uses air to remove some of the coolant's energy to disperse it into a moving airstream, similar to a radiator. However, the heater core's primary job is to heat the air passing through it instead of cooling the fluid in its tubes. Should the core develop a leak, disconnecting it is the best way to prevent water damage until you can replace or repair it.
The first and most obvious consequence of disconnecting your car's heater core is that the heater won't work. No coolant going into the core means no hot air coming from the vents.
Video of the Day
Disconnect the lines going to and from the heater core, start the engine and the reasons for a bypass tube become immediately obvious. The water pump will continue to send a steady stream of coolant to the core, which will spray all over the engine bay. The quick solution is to insert a piece of metal tubing into the ends of the coolant input/output tubes, joining them together to create a large bypass tube. You could zip-tie the tubes to your engine, but you might consider just cutting the core output tube and connecting it directly to the water pump's heater core outlet.
Odds are fairly good that you've disconnected the heater core because it's developed a leak. If this is the case, then you're better off removing the core and repairing or replacing it than leaving it permanently disconnected. Repairing the core is probably cheaper than replacing it, and leaving a heater core sitting empty for long periods of time will allow corrosion to eat away at whatever's left of it. If, for some reason, you're disconnecting a heater core that hasn't developed a leak, then fill it with anti-freeze and cap off the ends with a spare piece of tubing to prevent corrosion.
Racers often opt to remove their heater cores and air conditioning systems in the name of saving weight. This is a perfectly valid mod if your car never leaves the drag strip -- after all, a full heating/AC system can easily add 80 to 100 pounds to the front of the car -- but be realistic before stripping your car to a shell. Unlike the heavy A/C system, a coolant-filled heater core and blower motor only add about 15 to 20 pounds to the car, and it doesn't pull any power from the engine. Even central Florida has its share of 20-degree nights, and that 0.01-second you'll shave off your 1/4-mile time won't seem worth it when you realize you can't shift because you can't feel your fingers.