I'd be worried - can't see how a solid state LED would produce vibration, but electrical leakage can feel like vibration.

Vibrations are usually caused by magnetics: chokes (inductors) and transformers. Hopefully it isn't an insulation problem as suggested by Matth.

Thanks for taking the time to answer, both of you.

I did a similar test at home with a different set of Christmas LEDs on 110V, and the electrical leakage hypothesis seems a plausible idea.
Everytime I turned the lights on, I felt a barely noticeable sting, then a very light "hum" in my hand.
And when I turned the lights off, another very small sting, then nothing.

I also tried this with my eyes closed and someone else turning the lights on/off. Same result : I feel the difference when the lights are on/off.

So if the electrical leakage is really happening, any idea if it's dangerous? (fire hazard, etc.)

It could be an indicator of bad things. Insufficient wiring insulation, incorrect isolation design if it's fed through a control or plug pack. If this happens on appliances or similar electrical apparatus, it's an indication of inadequate bonding to ground. Try plugging it into an RCD or GFCI outlet (in your bathroom or kitchen). I almost guarantee it trips.