Gas furnance won't stay lit.

Then that would be the limit not shutting off the limit works on temp rise on and temp fall off

make sure wiring is correct especially if wired 120 volt read instructions inside panel

Reliance water heater?
Thermocouple is good if the pilot stays lit continuously.

501 is older model. Unknown if this unit has FVIR protection and sealed combustion chamber or open air intake on front of unit.
http://waterheatertimer.org/pdf/Reliance-2006-Service-Handbook.pdf

With FVIR protection:
Get more fresh air to the unit.
Clean the flame arrestor screen.
Test and replace the thermocouple:
http://waterheatertimer.org/How-to-test-water-heater-thermocouple.html
Inspect and clean burner and combustion parts
Test gas pressure, bleed air out of gas line.
Replace gas control valve.
Test draft air by lighting match and see if smoke is pulled into the vent pipe at draft hood on top of tank.

If you need further help, I’m available over the phone at https://www.6ya.com/expert/gene_9f0ef4df2f9897e7

i would start by checking the flame sensor ...if the unit is actually firing off just not staying lit sounds like a faulty or dirty flame sensor.... if you look inside at burner assembly there will be a metal rod ranging from 1 1/2 to 4 inches in length.... it ia in burner assembly and has (1) wire that connects to it and usually one 1/4 inch screw on it.... remove this and lightly /let me emphasize lightly sand it and clean it ...to much will actually damage it...re-install and see if it helps... this senses the flame and tells the control board to keep gas valve open most likely your culprit....good luck to you...

Gas valve is not kicking in or gas shut offs are off. Can you check voltage where 2 wires connect to gas valve. Can you see blinking lights thru lower door window and count blinks. Code sticker should be on inside of lower door.

you need a service person to come out

It is probably not the glow coil, if this is a 80% regular
furnace, Make sure the induced draft blower is working. I once went to a house where I came across
this problem, the only difference is that it would acutually start and then shut off. I took the vent apart
at the furnace and found a bird that was preety cooked.

If this is a90% High Eff. furnace, make sure the pvc
vents are not plugged. To troubleshoot the intake
Vent. Remove the combustion plate and then turn
the unit on, if it runs you have a problem with the
intake vent.

You could temp. remove the exhaust vent and then turn on the power if it runs then your exhaust vent
is probably plugged. You could go outside and make sure the vents are free.

Make sure you have gas available, these are the most commom issues that could solve your problem

Sounds like you need to take some sand cloth and clean the flame
sensor located on the far right side of the burner

. Several times of recyling power normally brings the furnance up. Though the failure code has always been 31, there are several different behaviour when the blower starts after reset :
a. blower simply creases.
b. there is an attempt to start the heating unit (I hear the high pitch noises), then everything stops.
c. there is an attempt to start the heating unit (I hear the high pitch noises), once I see the orange glow from the coil, normally the gas is lit up and the unit starts working.

There is a vacuum sensor on most units tied from the exhaust gas fan to the sensor. Does the unit try to start? If the unit's exhaust fails to pull the switch it will turn off after about 30-45 sec after starting. "IT" thinks it's not exhausting and stops to keep from overheating. Do you have this exhaust fan and rubber hose to sensor? Can you see if that fan is turning on start?