Electrical and Computer Engineering

The LED doesn't make any noise. The only thing you could be hearing is a faint whine from the power supply current changing, although I'm surprised that an LED would have enough of an effect to noticeably change the sound.

Removing the LED will not damage anything. No reputable product would be designed so poorly that a small decrease in current would break something. Depending on where the LED is though, breaking/disconnecting it could be far more trouble than it's worth, and you might risk damaging other parts of the monitor trying to get to it.

I would agree with the other commenter, your effort is probably better spent on something to help you remember to turn the monitor off. Or, even just buy a smart outlet and plug your monitor into that, so that if you forget, you can use your phone to power off the monitor without getting out of bed.

Get a smart plug that you can control from your phone?

Bonus points if you install Home Assistant on a Raspberry Pi and configure the plug to turn on and off automatically based on the sunset time in your area combined with the moon phase....

Just to make sure... Did you check whether the LED can be turned off in the monitor settings?

If that is not the case, I would suggest getting a smart plug, and specifically one that can monitor the power consumption. That way you can automate it in such a way that it switches the power off when the power consumption drops below a certain value and stays there for a bit. I set this up in multiple locations in the house and it helps save on standby power usage. I use node-red to monitor and control the plugs, but other options like Home Assistant would work as well.

The plugs (Shelly) I use even have the option to set up the automation in the app (which requires cloud connection to be enabled), or run scripts on the devices themselves (no cloud required, but not all their devices support local scripts. You typically need the newer "plus" devices for that. You can then easily set up an automation that switches the plug off if it is powered on and the power consumption stays below 10W for 5 minutes for example. You could put your whole desk on that plug, which should save you some money on your energy bill as an added bonus.

I made a script which turn my monitors off by sending a couple of DDC commands via ssh from termux so I can do this from my phone while in bed. Via ddc you can turn them off like the button press, not sleep

Do the lazy and safe thing; Electrical tape over the light.

I strongly recommend against it as you will probably break your monitor trying to get the sealed case open, but if you did get to the LED and cut it out, measure the resistance, and replace it with a resistor of similar resistance, that should nip any changes to current flow in the bud. Again, I stress, the most likely thing that happens here is you break your monitor trying to open it.

Personally, I recommend you just set a reminder to turn it off before you go to sleep.

Just put a bit of sticky tack on it. If you need it to blend in, paint the putty out use black marker.

Then it's still removable if you want to sell it later