Wow! What a lot of really good, specific, detailed advice. I haven't gone to the house to inspect and try to resolve the problem yet. Probably tomorrow.
Here are my responses to some of the issues which have been raised here:
1. It's a Williams natural gas wall furnace, an exact replacement for the one which had been there since the 1950's. Using a replacement which was virtually identical to the old one made the installation much easier than it could have been. I had never installed a furnace before, so ease of installation was important to me.
2. The renter held down the pilot knob for a full minute---twice as long as the instructions recommended. My guess is that she held it down for 30 seconds the first time she tried it, and when that didn't work, she held it down longer. (She's a great renter, and if I told her I wasn't going to be able to get out to fix the furnace until July, she would probably find some way to deal with it.)
3. Yes, only a one year warranty. That's what the instruction manual said. A different brand of furnace might have come with a longer warranty, or a paid installer might have offered to stand behind it for longer than a year. But I wanted to use this brand and model for the reasons I explained above, and I saved over $1000 (nearer to $2000 compared to the highest bid) by buying and installing it myself.
4. It is a standing pilot model---you light the pilot with a match. I'm glad the renter reminded me of that or I might have gone there without matches. The natural gas freestanding stove I have in my house lights with a piezo-electric striker (I think that's what it's called), and I had gotten it in my mind that the pilot in the furnace in the rental house lit the same way.
Thanks, everyone. I feel much more confident about dealing with this now. I'll post the results when I get some.