Furnace pilot light will not stay lit after releasing and turning to on position. ?

As Bruce said it probably is a weak thermocouple. That is if it has a proper flame that is actually hitting the thermocouple properly. If not it cannot generate the voltage needed even if the thermocouple were good. A dirty pilot may not allow the flame to come in contact with it.

Probably the thermocouple. It is the tube that is mounted in the middle of the pilot flame. It generates a very small voltage that holds the gas valve open. It is an inexpensive part and relatively easy to change. But, before you change it, follow the tube or wires that lead from it to the gas valve, where it connects to the vale, check the connection(s). A loose connection there could be the problem.

More Info- Thermocouples come in two basic types, the first is a copper tube, 1/4"-3/8" dia, x 3" long. it has a thin, flexible copper tube coming from the bottom, with a connector at the end that looks like a plumbing fitting. the second is called a "millivolt generator". Instead of a copper tube, it has 2 wires, with electrical connectors on the ends. They both do the same thing, supplying voltage to the gas valve, as long as the pilot is lit. If you would like to test the thermo before you replace it, you'll need a volt meter capable of reading millivolts. Disconnect the connector(s) and, with the pilot lit, measure the voltage, it should be 30 millivolts. The generator is straightforward, the thermocouple is more challlenging. At the end of the tube is an outer conductor, seperated from an inner conductor by a thin insulator. It's a little tricky, but measurements can be taken.

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Most you have to hold down after it is lit for a full minute before they stay lit. If it still doesn't work you may have a bad relay, or valve (a safety device built into your stove.

You didn't say what type? Oven, heater, even the model number alot of things epically stoves you can find a manuel on line even if its 20 years old, according to where it was made in.