LED glows slightly during soldering
Although LEDs ought to include a reverse diode for ESD protection, the heater coil induces a magnetic field into the high impedance LED with enough mutual coupling to generate xxx µA of dim current.
Anecdotal
I sold over 1 million 5 mm LEDs, and it wasn’t until I flew down to this one client in New Zealand and saw their poor handling and soldering practices that I changed the design of the LED to include ESD protection and a dozen recommendations to improve their volume manufacturing process. Then their 0.5% failure rate dropped to near zero.
Although the current may be low, the reverse voltage may exceed -5 V and partially wound the junction.
It is probably a small electric current from the tip of the soldering iron. Some irons have a grounded tip, others may not. In both cases, it is possible to have some voltage at the tip.
You can try to test for this with a multimeter set to AC volts from the metal body of the soldering iron to the LED, but a meter might not have enough sensitivity at about 10kΩ impedance. A better tool would be an oscilloscope with 10MΩ impedance. The iron may need to be "on" for accurate results.
For some reason, the electrical outlet in my old kitchen had about 69vAC on the ground, but only at a very tiny current, perhaps 20µA, which a meter would barely detect. This was tiny, but it was enough to "light" LED's for no apparent reason when touched with the iron. Note this made it unsafe to solder more sensitive components, such as digital IC's.
I had a suspicion this was related to an old refrigerator. It had a propensity for shocking you slightly if bumped with a wet arm. Unplugged it and sure enough, the leaked voltage at the nearby outlet disappeared.
Don't fool around with this! Test it.
Seek out a grounded, GFCI-protected outlet (kitchen, bathroom) or RCD-protected outlet, preferably a 6 or 8 milliamp rated one.
Remove the cover plate from the outlet*, so you have access to the yoke (metal wings of the receptacle which take the mounting screws). This yoke should be tied to the Equipment Safety Ground,** which bypasses the GFCI. We need access to that.
Now, plug the soldering iron in, and touch it to the yoke repeatedly as it warms up. Don't hold it there, or you could melt the receptacle or plastic junction box.
When that's done, if it's possible to flip the plug over (looking at you Germany), flip the plug over and test again.
If it trips the GFCI, into the trash it goes
If the soldering iron is leaking current to the tip, it will trip the GFCI in about 10 milliseconds. That is an indication to throw it out, because it is internally leaking mains current onto the tip - which means it will leak mains current onto you. 10ma can kill. It can also stun you, and unconscious person + soldering iron is a bad combination.
If it is an expensive unit, you can maybe do a teardown to see if you can find dirt, splashed solder or frayed insulation that might be causing this.
What is the test testing? The GFCI detects by comparing current flow on hot and on neutral, to assure they are the same. If they are not, some current is taking a third path. So it detects all current flow that is not hot->neutral. We are using ground to create a third path for testing. We flip the plug over to exchange hot and neutral, because neutral->ground is harder to detect than hot->ground.
* As a low voltage DC electronics person, if you normally have a healthy fear of all things mains power, that is a good thing. And if you don't have that fear, obtain it. LVDC does nothing to prepare you for the hazards, rule-set and mind-set of mains power. But you can certainly do this.
** No relation to electronics GND/VSS/"common", that would be neutral. Equipment Safety Ground is a shield, but does not contact any mains conductor in any way (except for exactly one equipotential bond). As such, it entirely bypasses the GFCI - really, GFCIs don't use ground (except GFCI receptacles, to pass it through to appliances).