A thermostat that is calling for heat or cooling while the room does not change is showing you a signal that stops somewhere between the control and the equipment. Your job is to find the stop before you assume the equipment has failed.
Confirm the call is actually being made
Check that the thermostat is set to the mode you expect, heat or cool, and that the set point is far enough above or below the room temperature to call. Look at the display for a call indicator if it has one. If the thermostat runs on batteries, check whether the display is dim or blank, since low power can leave the call unclear.
Check the equipment's own switches
Find the power switch at the unit, the service switch near it, and the breaker for the heating or cooling equipment. Look for a panel cover that is not fully seated, since many units will not respond with the cover out of position. A switch in the off position can stop a call from reaching the equipment even when the thermostat is working.
Watch what the equipment does
Note whether the blower or fan runs when the thermostat calls. If the fan runs but the heat or cooling does not, the signal is reaching the unit and the question moves inside it. If nothing runs at all, the wiring between the thermostat and the unit is the next stop. Log what you see with the time and the setting you used.
Know where to stop
Gas burners, refrigerant circuits, and live electrical parts belong to a qualified technician. Your part is to confirm the call, confirm the switches, and describe what the equipment did. Bring that log to the visit. For the gas side, the Gas Furnace: typical service life and when to plan page sets out the planning date that goes with the unit's install year.