Recently, every time hot water is turned on anywhere in the house, water trickles from the faucet in the shower. Where should I start in trying to fix this?

You are saying from the tub/shower valve spout, right? It drips for several minutes and then stops I'll bet.

Let's get together on terms. Showers don't have faucets. Tubs do. Sinks do. But not showers. So that leaves us guessing or asking for clarification like Bob's douing.
How about being more specific. Don't make us guess, It's hard on the brain. Cheers, Tom

Is the tub or shower or tub/shower a single handle or two handle or three handle. Im guesing it's a single handle If so, I would start by replacing the cartridge, as the tempering device in the cartridge is most likely malfuntioning.