I have a slightly annoying habit to turn off electronic devices whenever possible, but I would like to use the Pi Zero W as a lightweight web server for personal use. Any advice to get over this habit?
Yes, it is safe to leave any Raspberry Pi on 24/7, including the Pi Zero W. "Safe" in that case means no damage to the Pi. For the environment to be safe, e.g. not burning down your house, the Pi should be in an insulating case and the power supply should be of high quality. (I doubt you could really start a fire by short-circuiting the power supply, but you get the idea.)
Finally, there's software safety. My home gets a power "blip" every month or so and a failure of minutes or longer every couple of years. Sudden removal of power can corrupt the microSD card, so have a backup. If the service provided by the Pi is critical, consider an uninterruptible power supply.
Here are some tips to help achieving a longer lifetime of your Pi:
- Use a well suitable PSU (really recommending the Original Raspberry Pi Power supply here)
- Place the Pi in a well ventilated area, a fan should not be necessary
- Secure it sufficiently (e.g. turning off passwort authentication and using a SSH key instead) so that it is harder to hack
- If the power drops out often, I suggest you to use a powerbank or a UPS board
Then it should be safe to leave it on all the time, but you might want to get a USB-to-Ethernet adapter or an ethernet HAT/shield for faster and more reliable network access.