Yeah, Mike's post brings up some good possibilities. If it isn't the cap gasket or pip, varnish may well be causing the needle to not shut off the stove properly and allowing leakage there through the jet. I think that about the only other places are likely to leak are where the burner screws onto the tank or the packing.
I think I'd either try take the cap off or the pip assembly out and rig a way or pressurizing it via the fill opening to a low pressure or take it apart and clean up everything before reassembly. Not sure how much pressure is okay so be cautious, it shouldn't take much to identify the leak. A spray bottle with soapy water ought to be able to help spot the leakage if you do the former. That or just submerge it in water while pressurized.
Heck you could probably just submerge the whole stove in very hot water without opening anything. That ought to pressurize it enough to make air blow out a tiny stream of bubbles the leak point. I'd drain it of fuel first and rinse it out with Heet after to get rid of any water that might have gotten in. The Heet will also serve as a varnish cleaning. I think you want the yellow bottle Heet not the red.
My preference would probably be to avoid taking it apart and rebuilding unless necessary, but some might prefer to do a full rebuild with new wicks, packing, and rubber parts polishing everything to a shiny new brass look in the process. I'd rather wait until my stove is old to rebuild it
After all I only bought it (used) in the early 1970s and it was probably only 10 years old then. It is still practically new at probably less than 60 years old. Seriously unless you cook them dry and scorch the wicks or burn bad fuel they usually just keep going and going.
Most likely your stove has some simple minimal problem that once identified is easily fixed.
