There's a leak, most likely a freeze burst, in a vertical section of 1/2" copper pipe that runs behind some drywall. In order to minimize the amount of drywall that I have to tear out, I'd like to locate the burst before hacking up the wall. Is there a way to do this?
Sure, just turn the water back on and wait.
Of course, that won't actually minimize the drywall repair work. But you'll eventually localize where the leak is.
My actual approach would be to point out that a big drywall repair is barely any harder than a small one, so get over that misconception and rip away.
If you are really dedicated to avoiding replacement of a sheet of drywall (see other answers), then drill a small hole near the top of the run, and insert an endoscope or equivalent. Lower it slowly until you see where the drip is occurring. Cut there.
You can get USB-powered & connected scopes like this for peanuts at the usual shopping places online.
If the leak has been going on for a while and is bad enough to allow water to spray out sideways so it hits the drywall (instead of just dripping out slowly and traveling down the pipe) you just need to gently but firmly probe the drywall with your fingers until you find a soft spot caused by the drywall absorbing the water. Since it is already leaking there, your drywall is getting wet and will almost certainly grow mold and mildew, so you will need to replace a it anyway. If the leak is spraying out at an angle so you're hole isn't exactly at the leak, it won't matter since you needed to patch that spot anyway, but you should also be able to look through that hole to determine whether you need to move up or down.