Will a magnet stuck in my front loader wreck the door lock?

The door almost certainly isn't broken, it's the magnet. A lot of front loaders use a magnetic sensor to detect wether the door is latched, so a strong magnet floating around in the drum can convince the machine the door state is something it doesn't like, and it just refuses to unlock. Same reason it stopped mid cycle and started beeping. The good news is these sensors usally go back to normal once the magnet is gone, so I wouldn't worry about long term damage yet.

The water in the drum is the other part of the problem. Most machines won't release the lock while they think there's still water in there, so draining it is step one regardless of the magnet. On most front loaders there's a small access panel at the bottom front, and behind it you'll find the drain pump filter. Some have a short hose with a cap next to it, which is the tidy way to do it. Either way, put down a bunch of towels and a shallow pan, because it'll come out faster than you expect. Unplug the machine before you open that panel, you'll be working near the pump.

Once the water's out, try the door again. If it still won't budge, a lot of these machines have a manual release tucked behind that same access panel, sometimes a pull tab down near the lock itself. Check your manual or look up your model number online, it takes two minutes to find.

After the door opens, fish out the wash cloth and the magnet, and then check that pump filter carefully. If a screw or bolt got past the magnet at any point, that filter is where it'll be sitting, and that's also the most likely thing to cause drain problems later. Get any loose metal out before you run another cycle, because it can chip the door glass or scratch the drum.

As for the hack itself, tell him to keep a magnetic parts tray by the hamper and empty pockets straight onto it. Same result, and nothing ends up in the drum.

Had the same panic with a bra underwire that jammed my lock a few years back. What got mine open was the drain first, then the little pull tab behind the bottom panel, took me two tries to acutally reach it. Also, since you can hear the magnet clunking around, try tipping the machine back gently and sliding it toward the door with the drum tilted. Mine moved right to the opening that way once the lock let go. After you get it out, run an empty rinse and spin and watch it drain. If the pump is quiet and the water goes, you're done. Mine has been fine ever since.

Following this because I'm nosy and want to know how it turns out. One thing I didn't see addressed yet is the wet clothes. If this drags on a couple days while you figure out the drain and the pull tab situation, that load of work shirts is going to sit there souring and you'll have to rewash everything anyway. Does the machine just pick the cycle back up where it left off once the door finally opens and closes again, or do you have to start the whole thing over from the beggining? I've always wondered that with mine.

Part time appliance tech here, I pull maybe two or three of these magnetic hacks out of machines every month. One thing to check once you get the door open: run the machine empty on a rinse and spin and watch if it drains properly. I had a customer whose rare earth magnet actually stuck to the drain pump housing itself, sat there for weeks, and the pump still worked fine afterward. These sensors are tougher than people think. The bigger risk is the magnet wedging against the door boot and tearing the rubber, so inspect that gasket all the way around before your next real load.

Update: I pulled off the bottom panel and found the drain hose, got most of the water out into a shallow pan, and tried the little pull tab the underwire person mentioned. Still nothing, door acts welded shut. The tech's comment about a magnet sticking near the drain pump has me wondering now. If it's stuck somewhere inside, how do I even reach it without pulling the whole front off?