A few years after adding onto our house and some settling had occurred, we started hearing this strange noise in the closet wall behind my games and in the heating duct in the floor in our bedroom (directly above the game cabinet). The noise only occurred when the water or shower was turned on in the upstairs bathroom (above the closet). It sounded like dripping (sort of) coupled with the heating duct bonking like metal warping one direction and then another a few times. The sounds would last about 30-45 seconds and then stop. Turning the water off and on didn't make it come back and you'd have to wait a significant amount of time for it to happen again (hours). It was worse in the winter.
After some investigation and trials I determined that it was only when the hot water was turned on. It's slowly been getting worse and it's started to drive me nuts. So, I decided to clean out the closet and cut through the drywall to expose what was going on back there. Luckily, when we built the house I had taken lots of photos of the walls after plumbing and HVAC were installed but before drywall so I knew where everything was.
After opening up this hole in the closet wall, a hole in the ceiling and a hole farther up the wall next to the ceiling I found my problems.
The first problem was that settling had allowed the hot water pipe to be forced against the ductwork feeding a bedroom register. The expansion of the hot water pipe was rubbing against the duct.
My second problem was that the pipe was being forced hard against the side of the hole drilled through the ends of three studs heading the wall in the closet. Again, the expansion of the pipe was causing minute rubbing against a knot in the 2x4s making the tiny "drip drip" sounds. Once the pipe expanded fully the noises would stop and only after cooling down and contracting would it happen again.
So, I exposed the ends of the header studs to free up the pipe and that allowed it to move away from the duct about 1/8". I went ahead and placed an insulated foam pad between the pipe and the duct for good measure. I'm now in the process of patching all of the drywall but I'm a happy camper. No major problems, no major issues, no trips to the hardware store, and no money needed for the fixes.