Replace manual thermostat with programmable
You may not be able to.  The programmable thermostats commonly sold are designed for on/off control of a heating and cooling system.  The Flair line of thermostats are designed to control a Flair damper that is allowing more or less air into the room.  The Open and Closed Signals move the damper in that direction.  I believe the Flair is an variable system, not the on/off a thermostat is designed to control.

You could try an old hack from the days before programmable thermostats.  Mount a 4 watt incandescent nightlight directly below the Flair thermostat and wired into a light timer (or a home automation module).  When the nightlight is on the thermostat will be fooled into thinking the room is a few degrees warmer than it actually is and the real temperature in the room will be "set back" several degrees cooler.

If you have some electronics skill a SPDT relay could have its contacts wired to the Flair wiring.  (Closed to the NC relay terminal, Open to the NO terminal, and common to common.)  A low voltage power supply (under 24V) that matches the coil voltage of the relay could be wired to the could be wired into the R terminal of the thermostat, and then out via either the RH terminal (heat) or Y (cool) and to the coil, then back the the power supply.  This won't work if the thermostat is designed to be powered from the furnace instead of having its own batteries.