Can I keep a Linux laptop running with the lid closed and screen off?

First separate the two jobs: lid handling controls whether the computer suspends, while the display’s backlight may turn off automatically when the lid closes. On systemd systems, check the lid-switch settings in logind and set the relevant handling to ignore if you want the machine to stay awake. Then test that the screen actually goes dark on your model.

Be aware that ignoring the lid switch means the laptop can keep running in a bag or somewhere with poor airflow. A server sitting open on a desk is a different situation.

GNOME’s power settings are worth checking, but they may not offer a separate “turn off the panel, keep everything else running” option for every setup. If the hardware already cuts the backlight when the lid closes, changing the lid action may be all you need. If it does not, the exact display-off method depends on the graphics stack and session, so don’t assume an X11 display command will work under Wayland.

For a machine that mostly serves files or media, you could also run it without a full desktop environment. That avoids some desktop power-management behavior, though you still need to configure the system’s lid action and make sure it stays cool. If you want a graphical desktop for remote access, keeping it installed is reasonable; it just may take a little more testing.