Sounds like the GFCI receptacle got wet (or perhaps a downstream receptacle if there is one), and something is shorted out, tripping the GFCI.
If you have any Christmas lights still plugged into the outlet, then something may have shorted out in the lights, making the GFCI trip. Try unplugging everything and see if the GFCI will reset. If it does, then dry out and inspect the lights carefully to look for problems -- don't plug them into a non-GFCI outlet since there is clearly an electrical problem and use care around them until the problem is found.
If it's not a problem with anything plugged in, then new GFCI's are cheap enough that I'd just try replacing the GFCI (fixing any water intrusion problems first, if there's water in the box, then figure out why and stop it from getting in there). If the bad GFCI eventually dries out and starts working again, I personally wouldn't trust it again.
This is likely not a serious fire hazard, but could be an electrical hazard (since something is apparently shorted out), so to be safe you could turn off the breaker that powers that outlet.
If the outlet got wet because the lights were plugged into the outlet while the outlet cover was open, you can purchase an outdoor outlet box that allows a cord to be plugged in even when the outlet cover is closed:
(this is just a representative sample, there are other styles and manufacturers of similar products)
My suspicion: an open ground
This is based on two factors:
Your three-lamp tester is unable to trip the GFCI, but the GFCI tests OK. This is because the internal TEST button doesn't rely on the ground (it goes load-hot to line-neutral or vice versa), but the three-lamp tester's TEST button is connected hot-to-ground, and thus requires a solid ground connection to work.
Pushing the TEST button on your tester yields a "hot/neutral reverse" indication despite it testing correctly initially -- the "correct" indication is due to capacitive coupling between neutral and ground providing a current path "home" for the current through the right-hand lamp.
I'd check to make sure the green wire is securely connected to the box in your case; if it is, then the problem's upstream somewhere as it appears this receptacle was wired using conduit (note the locknut and fitting at the bottom of the box). If you can't find it still -- call in an electrician, as they should have better tools (such as an earth fault analyzer/loop impedance tester) that can be applied to the problem.
Best Answer
The GFCI receptacle has power on the LINE side, and that is where the status lights are powered from. The GFCI is in tripped mode, which means no power comes through to the LOAD side, which includes the sockets on the face of the GFCI and anything attached to the LOAD screws.
Troubleshooting the GFCI
First, it's rarely the GFCI. But let's exclude it anyway. To do this, we must isolate the GFCI. That means we must remove all wires from the LOAD terminals of the GFCI - all of them, don't go thinking you can leave neutral.
Now, with absolutely nothing on LOAD, the GFCI should run through its paces. It should reset, and Test should trip it. The magic 8-ball tester's GFCI function should also trip it if it is grounded (otherwise it's supposed to not trip, don't try to modify the GFCI's wiring so it does).
Does the GFCI refuse to behave with nothing plugged in and nothing on LOAD? Then it is the GFCI. Otherwise, it is not, and never was.
If the GFCI isn't bad, the downline wiring is
People have a dreadful habit of using the LOAD terminals merely as extra screws to put on more wires. Never do this. If you don't understand what LOAD actually does, don't use it. If you don't actually intend to protect the downline, don't use it. Move those wires to LINE and leave the warning tape on LOAD.
However, it is a fact that your downline wiring has a problem. Moving it to LINE would mask that problem. So you may want to leave it on LOAD and actually fix the problem.