This is an easy fix,,,,,,,,,,,,,,,,,with a volt meter and knowing how to use it. Electrical troubleshooting isn't somthing to be taught on the net long distance. I can help you part ways though. Do'es the burner ignight but blower does'nt start? That is a motor standing on end isn't it? Is that the only motor or is there a secound farthur back lieing horizonal?

Thanks so much for your reply Klem. As far as I can tell, our heater has neither a blower nor a motor. When it works, it just ignites from the pilot. It is a "Lear Siegler gravity vented wall furnace," according to the plate inside. And I'm guessing it is pretty old. I'm not finding any info on the model number on the web.

Here's what the whole heater looks like:

Ok,I misstook the air inlet pipe for a motor housing. You are correct,no motor is involved.
Despite your dodging the volt meter,I will try to help you. And please,if anyone here on G.W. knows by looking at the pics what the problem is, spare mamalinda the hassel of testing.
First off,the symptoms point to a loose or broken wire or t-stat. You stated "I've replaced the batteries in the thermostat and looked at all the wiring - nothing seems to be loose or breaking." , that must be 100% true or our tests will be flawed.
One pair of the wires connected to the blue plastic is from t-stat ( we will call them T and Tc ) and the other pair go's to the pilot lite ( we will call them P and Pc ), We know that Tc and Pc are the 2 wires in the center, The wire on end that go's to pilot is called P which leaves the wire on opisite end as T.
Turn t-stat on and calling for heat. Touch one lead of volt meter/test light to either center screw and other lead to an end screw then other end screw.
Tell us:
A. If you got readings and which end screws had a reading ( YOUR RIGHT, LEFT or both)
B. If each pair of wires remain within the same outer jacket as they go from blue plastic to pilot and t-stat. Or conversly,if one wire within a jacket terminates at pilot and other terminates at t-stat.
When we have your answers,we will continue from there.

Note on meter, if needle deflects backwards when screws are touched, reverse leads because -- & + are reversed. A digital meter will simply read as a negitive voltage if leads are reversed.

Note on test light, note voltage on one of the batteries and multiply voltage by the number of batteries in t stat. That is the volt value your test light must be within 10% of.

I've got a message out to a friend to borrow a volt meter and I'll post the readings as soon as I get them.

Thank you so much for your time!

I've got a message out to a friend to borrow a volt meter and I'll post the readings as soon as I get them.

Thank you so much for your time!

Oops. Didn't mean to post that last one twice.

Ok. New development: I turned the pilot off a couple of days ago since the heater wasn't working anyway. So, after getting the volt meter from my friend, I just re-lit the pilot and turned the thermostat back on to follow the directions you posted. And the heater came on. And now each time I turn the thermostat down it goes off and when I turn it back up it ignites and runs just fine.

I'm sure this is not going to last - I imagine it will do what it did last time and run ok for a while and then stop igniting.

My question now is: do I need to wait for it to stop working again to test it with the volt meter? Or can I test it with the heater running? Or should I turn the pilot off again and test it with the pilot off?

Or, does the fact that it's working again now, after the pilot having been off for a couple of days, give any more clues about what's wrong with it?

Thank you again for your time.

The pilot had no effect on it. Since you have the meter,you may find the problem before it surfaces again. For what I'm suggesting a light or buzzer is easier for a novice but we will go with the meter. Just be aware that reading may only be momentary so be very observant in order to see it.
I believe the fact that pushing temp adj higher than normal to get it to operate strongly suggests intermitent contact due to A, Loose connection at t-stat B. A wire broken inside plastic insulating jacket C. Loose/broken part on t-stat D. Battery holder making poor contact with batteries.
Therfore we will start at t-stat. With heater flame on, have your helper hold meter probes on both screws at each end of valve. The reading should equal sum of battery voltage. If not,forget the meter. Lower temp at t-stat. Meter should drop to 0 , the valve should make an audiable click and flame should go off. Raise temp and meter shows voltage,valve clicks and flame resumes. Gentely move each wire at t-stat. Touch and wiggle batteries. If signal wasn't interpted,move wires at all connections inside furnace. If still no interuption,you will have to wait until the problem accures then test for voltage at valve. If valve has voltage but there is no flame,defective valve. If valve has no voltage,you missed a loose connection during the test.