Have a Dependable Ninety Two with Honeywell Thermostat, burners ignite and run then shuts down and starts again. The Thermostat does not shut off the burner cycles (heating cycle).

SOURCE: After a few heating cycles the burner does not ignite

The problem you are describing is usually associated with a bad gas valve coil. The dryer runs normally until it gets hot, then starts making a strange clicking noise. After it cools down, the dryer runs normally again. The gas valve coils (there are two of them) usually run about $20 a piece. I recommend replacing both of them at the same time just to be sure. I hope this helps.

SOURCE: Honeywell Gas Furnace

A Make and Model number of the furnace not the thermostat will help but I can walk you thru the operation of a typical furnace. It sounds to me that the burners are igniting properly but the flame sensor is not sensing the flame and shutting down the system. You will need to find the sensor and clean it and check the wire from it to the control board. Typically, the flame sensor is found at one side or the other of the burner box. It is a small probe that the flame hits. Depending upon your model it may have a small wire attached to it leading back to the control. Or it may be part of your auto-pilot assembly. ? At any rate, you will need to get to it and very delicately brush it off. Then inspect the flame channel. This is a trough area where the flame travels to ignite all burners. Over time, they tend to fill with rust and debris. Take a small screwdriver and run the blade down the trough. Inspect the wire that goes from the flame sensor to the control board. Make sure it is not frayed or broken or shorting out and that the connection at the board is good.

Beyond that, it's time for a technician to come.

Let us know the Make and Model of the furnace (found inside the furnace cabinet) and maybe we can help you further.

SOURCE: Pilot goes out after heating cycle

Check regulator on the unit.
Check pilot unit on the heater
use soapy water to check for leaks if see bubbles turn off main gas supply valve.
always turn off gas supply before any checks or repairs are made
when all else failes call your local gas provider to check heater and gas supply

SOURCE: Heater Burner ignites but main blower motor will not turn on.

it sound like the blower motor may be seizing up. try spining the blade by hand to see if it feels free or not. the motor could be trying to turn but can then tripping itself out on overload. (most motors have built in safety overloads which will trip during an overcurrant event. aka too many amps.) you may have to replace blower motor

SOURCE: dependable ninety two blower comes on no igniter glow gas flows

if you dont see the glow of the ignitor it is probably bad . you can check with a meter that has an ohms scale, unplug and test ignitor side OL would be a bad ignitor 45-110ohms ok.