When you light your gas heater, get it lit, hold the button for the required time, let go of the button and it goes out, you will either need to replace the gas hold on valve or the thermocouple.
I am assuming that a known good source of gas is hooked to the unit and the igniter is sparking when depressed. In addition to the igniter sparking, it also needs to be in the correct location. It should be located just behind the end of the gas burner about 3/8 ths of an inch and approximately 1/4 to 5/16 ths of an inch from the burner. If the igniter is too far forward or backward from where the gas exits the burner the heater will be hard, if not impossible to light. Adjust the position, carefully so that you do not break the igniter, with a pair of needle nose pliers. When properly located you should get a nice big spark with a bluish tint in the vacinity of the holes where the gas exits the burner. Make any adjustments with the unit unplugged and the gas turned off.
When you light the heater you need the gas to be turned on, but if it would lose its flame it needs to shut off automatically. This is a safety feature to assure that the unit will turn off when any problems are encountered.
The way that the shutoff is accomplished by Reddy Heat is to use a hold on thermocouple. The heat from the flame makes the thermocouple output between 18 and 28 millivolts which keeps the gas on. This can be checked with a DMM (digital voltmeter).
Turn off the gas supply. Remove the thermocouple from the gas valve. With the DMM set to the lowest millivolt setting place the black lead to the tip of the thermocouple and the red lead to the outer casing or ground. Relight the heater and if the thermocouple is good you should read between 18 and 28 millivolts.
In addition some units use a relay in conjunction with the thermocouple. You will have to determine if yours does. Remove the bottom base, if there is one, otherwise locate the relay. With the DMM set to ohms, measure the resistance of the relay between terminals 4 and 6. With no power you should read infinite, (open), and with power you should read less than 100 milliohms or a short. If the unit reads bad, open both ways, replace it.
With the gas valve being held open by the flame the heater will run until the gas runs out or it is shut off. If the flame goes out the thermocouple quits outputting power and the relay opens, shutting the unit down. Repair parts can be purchased at www.reddyparts.com
These units do not have many parts and these are the most likely causes of your units failure to continue running.
Thanks for using FixYa and for the great rating.