Pilot light not working in gas heater

A pilot light has a sensor on it to know that a flame source exists before allowing any gas to come on. Otherwise you could be dumping gas into your home. So yes, a pilot light should always stay on for a heater to be used. Gas valves should close when the pilot goes out, preventing the heater from working.

Pilot won't light ...

If we are talking about an OLD TIME "open flame"
PILOT...

There is a lighting sequence... Push & hold the button
put a MATCH to the gas and count to 30 (thirty seconds)...

Will USUALLY get the job done...

The HOLDING for 30 seconds... ALLOWS
time for the THERMOCOUPLE to warm...
assuring the flame is active and RAW GAS is
not escaping (important safety feature)...

MOST common cure for FAILURE to stay lit: REPLACE
that THERMOCOUPLE...

Still carried by most hardware stores... it is important
not to OVER-TIGHTEN the new replacement...

Ideally... you should MEASURE THE LENGHT and
make sure you are buying the right length/size...

Amazon example
Honeywell CQ100A1013 24 Inch Replacement Thermocouple for Gas Furnaces...

eBay example
Universal Thermocouple 24 works with Most Gas Furnaces water heaters...

I buy two...
(I have a SPARE IGNITER or Thermocouple for
every oven, water-heater... heater and patio heater
that keeps my wife happy).

Carnac the Magnificent

Turn the gas knob to the pilot setting, press it in and hold it. You should hear gas escaping while in this position. Do not let more than about 10 seconds of gas out at one time as it may pool. Light the pilot while holding the gas knob down for a minute. Release the gas knob (the pilot flame should stay lit) and turn it to the ON position. To make sure the gas heater is working properly, turn up the thermostat and see if the heater heats up. How To Light The Pilot Light On Gas Heater

Make sure your gas is not shut off.
Do you have any other gas appliances that are working?

If you need further help, I’m available over the phone at https://www.6ya.com/expert/gregg_27a4d257e54e7920

If it's a direct spark ignition type heater maybe that's how it's designed to.work .When called for heat pilot open allowing gas then ignitor spark and ignite the pilot then gas valve open and allow gas through burner then unit fires off .
Once temp is met pilot valve closes and pilot out .

You probably have a defective thermocouple. The thermocouple sits in the flame of the pilot, and the heat generates a small electrical current conducted back to the gas valve, keeping the pilot gas line open after you switch it to the "on" position. That's why you have to manually hold the gas valve depressed (on) during lighting in the "pilot" position, allowing the thermocouple time to heat up enough to generate its current. A defective thermocouple will not generate the holding current.

If I understand correctly, the pilot light is not staying lit.
You light the pilot and water in the tank reaches temperature.
Once hot water is gone, then pilot light has to be lit again.

Gas pressure is good, and gas is flowing.
Thermostat is working at high end because it cuts off heating.
Vent is working because tank heats up.
Air intake is working.

Gas control thermostat might not be reading low temperature. Replace gas control valve.
Pilot tube might be dirty. Clean burner parts and combustion chamber, and air intake, and vent.
Backdraft might be blowing out pilot. Check for constant updraft by running water heater 5 minutes, and then light match under vent hood and see if smoke goes up.
Very hot air can put out pilot, especially when water heater located in attic. Add ventilation.
Improper vent might have water dripping back down into combustion chamber and putting out pilot light and rusting out the parts. Inspect for rusted parts and water.
http://waterheatertimer.org/How-to-install-gas-water-heater.html#troubleshoot
http://waterheatertimer.org/pdf/Pilot-light-will-not-stay-lit.pdf

There should be instructions on the inside of the heater access door. Make sure you are comfortable working with gas heaters. Make sure the shutoff valve supplying gas to the heater is on. Turn the gas valve to "pilot", press and hold it while attempting to light the pilot. Once the pilot is lit, continue holding the gas valve knob down for about 30 seconds. The pilot should stay lit once you release the knob. If the pilot stay lit, turn the gas valve to "on". Replace the heater access door and make sure you have proper water flow and attempt to light the heater. If it will not light, or the pilot will not stay on, you will probably have to replace the pilot generator. It provides voltage to run the heater and after time it wears out and produces less and less voltage. A millivolt heater runs on less than one volt so any power loss can be a problem. If you are able, test the pilot generator with a multimeter to see if you are getting at least 500mv when the pilot is on. If not, you should replace the pilot generator or even the entire pilot assembly. Hope this helps...

Could be moisture in the gas supply line - get your gas fitter to purge the line at the meter.