Are the large capacitors ceramic, tan, or electrolytic? If not ceramic, then leakage could be an issue. The voltmeter has an input impedance of 10 meg ohms. I was assuming that this resistance connected between the "node" and ground was enough to bias it back into the active region.
What happens if you connect the voltmeter between VCC and the "node" does it work? If it works, disconnect the negative lead of the voltmeter and touch the "node". Does it still work with only one lead connected?
If it only works when the negative lead is touching ground, then I suspect a pure dc bias problem. If it works with the negative lead disconnected, then I suspect that it is working only because the lead is injecting some power line hum.
Is the board surface mount or leaded components? Surface mount can be very difficult to get the solder flux out from under. Surface mount resistors and capacitors also can develop cracks that are hard to discover. When removing the part and checking with a meter, you apply an inward force to make contact. This has fooled me before by closing the crack and making the part test good. I would start replacing resistors a couple at a time.
After you heated the board, how long did it work? Cracked traces or cracked components will cool pretty quickly and return to a non-working condition. If it works until you power cycle the board, then perhaps you some sort of latch-up condition at startup.
Does it immediately stop working when you stop touching the "node"? If it again keeps working until you power cycle, then a latch-up is suspected.
My guess is that we are only looking at a small sub-circuit of the entire design. If this is true, can you break the circuit before this stage and inject the signal from another board? You can then take the input signal from this defective board and inject it into the known good board. If the problem switches boards you know that the problem is up-stream of the point at which you tapped into the circuit.