Normally, the word "furnace" is used for a forced air heating system, and "boiler" is use for a hot water heating system.

I'm presuming that by "furnace" you mean a "boiler".

If the water leaks a little bit and then stops, it could just be condensate dripping down from the chimney.

If it drips continuously, and it's coming from the middle part of the boiler, it might be a leak from between two segments of the boiler. If that's where it's coming from, then you're gonna need a heating contractor to take the boiler apart, fix the leak and put it back together. See if you can stick a mirror under the boiler and look around in there to identify the source of the leak. If I were you, I'd find out which of the heating contractors in your are deal with Burnham. They would have the most experience with this kind of boiler and know whether this kind of a leak is common with Burnham boilers, and if it can be fixed without taking the boiler apart.

In the mean time, maybe get a heating contractor that deals with Burnham to see if he can find the source of the leak. You might be best off to wait until summer to have it fixed.