120 volt timer passes small amount of current when off
This is a common issue with LED lights, and we have a few questions about in regards to normal light bulbs, but nothing I remember that is specifically about Christmas lights.

Plug in timers (and some smart switches, motion detector switches, lighted switches, etc) need to get power from somewhere so they can operate.  In the past, these devices have used the fact that incandescent bulbs need a ton of power to cheat the system a little bit.  Even when the device is off, it's running a very small current through the lights so it can power its internal circuits.

The problem is that LEDs will glow with even a minuscule amount of current, so this little trick that was invisible in the past, now shows up as glowing LEDs.  For Christmas lights, I would say it's no big deal, and just live with it.  It's not going to cause a problem.  They are cheap timers that are using a "vampire" method to steal power from the circuit.

If it's unacceptable, the only fix is to get a timer that says it will work with LEDs or a timer that is obviously powered by batteries.  These will use AAA or AA batteries.  A timer that only uses a coin cell is probably just using that battery for a timer backup and might still steal power from the lights.