I am having problems with one zone of a two-zone heating system. The thermostat does not function; previously this happened intermittently, and turning on the valve manually while the thermostat was set high enough started the furnace. Now, turning on the valve does not help either; the only way to get heat into this zone is to open the valve manually and turn up the thermostat in the other zone so that the furnace starts.

Should the valve opening turn on the furnace independently of the thermostat - i.e. is this clearly a valve or valve wiring problem? Or could it be a thermostat problem?

What brand of valve and model number?
Some valves have end switches etc so we need the model

The valve is by Honeywell: # 40004850-001 (only number I see on it)

V8043 or something like that is what you are looking for. The number you gave might be a number for a replacement power head which really does not answer the question but we are a little closer.

The end switch that is in your valve might be defective. The end switch is what turns on the boiler circulator pump when the valve opens. Now if the valve is not opening to start with there are a few things to check.

If the system is 24 volts or another voltage you need to test for that voltage at the valve when the system is set to turn on the valve. If voltage is missing it could be a transformer or other voltage source including the thermostat.

If when the valve is set to open manually the boiler/pump does not kick on you need to check the end switch contacts.

See this to check if this is the type of valve you have and the associated wiring.

http://customer.honeywell.com/techlit/pdf/69-0000s/69-0404.pdf

And here

http://customer.honeywell.com/techlit/pdf/60-0000s/60-2133.pdf

NOTE there are many configurations of voltage and switch as you will see in all the drawings and hook ups. Something should be there for your situation.

Well, I jiggled the valve box, took off the cover, tightened all the electrical connections and it works again. This happens periodically; I replaced the valve once already (a couple of years ago). Very mysterious... it certainly appears that it's not a problem with the thermostat, however.