Water dripping into fireplace?

We have a perplexing water leak situation.
It rained a lot here two days ago on a relatively warm day. Started getting drips both in our wood-burning fireplace and around it (i.e., coming outside the fireplace and dripping down the mantel). Assumed it was a chimney/roof flashing problem. Here’s the weird part: The weather has dried up, and it’s now much colder (below freezing)—and the dripping has increased! The plumber says it’s not a leaking pipe, and when we turned off the water to the house, the dripping didn’t stop. The water is cold, so I doubt it’s from our boiler.
Help?