Can dehumidification be achieved in a typical US residential forced-air system without dedicated components?
I will give you an off-label hypothetical answer for your hypothetical question TPE.  Do with it what you will.

Many medium efficiency systems have the evaporator coil placed downstream in the airflow after the gas furnace heat exchanger to insure that condensation from the AC does not form on the furnace heat exchanger, rusting it out.  If you have a system that has the evaporator downstream of the furnace then as Tester101 said, “nothing good” would come of running the AC and the heater simultaneously.  The furnace would overpower the evaporator and no moisture would be removed from the air, and you would have very big utility bill and your compressor might keep tripping on its overload protection.

If you happen to have a system where the evaporator coil is upstream in the airflow from the heat exchanger for the furnace and you are an ace at wiring your own control systems, then what you ask might be possible provided the following conditions are met:


The AC system should have a “low ambient” control installed.   This usually includes two components (i) a head pressure control that slows the outdoor condenser fan to keep head pressure above the minimum acceptable level when the outdoor ambient is below mid 70s and, (ii) a freeze-stat that senses the temperature of the suction line leaving the evaporator coil which shuts the compressor off for a few minutes when the temperature falls to freezing, insuring that the evaporator coil never  accumulates ice when it is being asked to cool already cool 68 degree air from the house.
The airflow for your furnace blower should be set so that the air temperature leaving your evaporator coil is in the mid to upper 40F degree range.  (You want to remove latent heat from humid 70F air, so you must get the evaporator coil well below the dew point).  To do this without risking early destruction of your compressor, your AC system preferably needs to have a thermal expansion valve (TEV) to regulate evaporator coil and not a fixed expansion orifice.  If it has a fixed expansion orifice and no accumulator to protect the compressor, then a skilled refrigeration technician will need to carefully set the system refrigerant charge so you do not get liquid flood back going into your compressor during cold operating temperatures.  Of course, you could have that tech retrofit your system with a TEV instead.
During dehumidification operation, your furnace would need to cycle to maintain the desired discharge temperature around 72 degrees F while the system is running.   The furnace is typically about twice as powerful as the AC in terms of BTU/hr, so it could not run continuously during the dehumidifying cycle unless you want to heat your house.
You would need to build or find a suitable mode switch for your system that would allow you to select between “HEAT, COOL, FAN, and DEHUMIDIFY.”
You would need to periodically inspect your furnace heat exchanger to make certain that if it forms an excessive amount of corrosion, you would know to replace it.