The burner lights up when it shouldn't

You are looking for a mechanical problem for an electrical issue. This unit is controlled by an electronic board. That board is the cause of your issue. They are about 300 dollars.

The pilot lights then heats/sends a signal to open gas for burner then(burner)should stay on but if valve shuts the gas off,then have your unit checked. What ever you do,DON'T MESS WITH IT IF GAS/FLAME ON!!!!!. These systems come with a safety feature that allows gas to go on only when safe!!!!

Try cleaning the flame sensor rod it will be in front of one of the burner tubes.

What are you expecting to happen?

I assume the water heater and water inside it is cold (otherwise you would not be lighting it). You light the pilot. The main burner comes on and the pilot lights it. The main burner should stay on until the tank is hot, When the tank gets hot the main burner should shut off but the pilot stay on. If you draw hot water, cold water will run in and cool the tank. If yu do nothing the tank will eventually just cools off. then the main burner should come back on to reheat the tank.

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

This is a great unit! Try turning the flow nob to the half way point and the flame all the way to low. Run the hot water tap and slowly turn the flame dial until you hear the unit try to light. Every house is different regarding the restrictions in the plumbing and water pressure. Once the unit tries to light that is your minimum setting. It is best to leave it there.

Ok the way this system works is from a pressure and flow switch. The pressure switch picks up the slight drop in pressure when the hot side water is turned on then turns on the burner. The flow switch measures the amount of water flowing and adjusts the size of the burner to the amount of water flow. So a couple different things might be happening. The flow switch might be sticking causing the water heater to throttle down the burner but not off when the water flow is off. Or the gas control valve might be by-passing gas when off causing the burner to stay on even when the demand is off. Run the hot water on high demand then turn the water off. If the burner is still on try banging on the water heater with your hand and see if the burner shuts off completely. If it does the gas control valve might be by-passing. You would also need to check and see if the pressure/flow switch is giving any voltage signal changes when the water flow is off. These tankless water heaters are more comlplicated to work on than a standard water heater and you may need a professional's help...good luck

First Clean the ignitor assembly and make sure the flame is reaching the flame sensor strongly.

You may need to replace the flame sensor.