should servers be left on overnight?

It is always recommended that the servers be on all the time.

Absolutely leave them on all the time.

Some BIOS manufacturers allow you to specify a time to power on, so I suppose if you really didn't have a use for the server overnight, you could easily schedule a shutdown time (say 8pm) and configure the BIOS to power back on at 6am.

Kudos for reducing your carbon footprint :)

Strictly speaking, do they have to be?  No.  But they are EXPECTED to be as many tasks such as backups and updates occur at scheduled times that the server is not expected to be doing much of anything (like 3am).  

Furthermore, if the server is your mail or web server, turning it off would result in delayed or lost e-mail and your web site being unavailable.

Plus, if you learn to exploit the server appropriately, you can use it remotely while at home, so turning it off would create inconveniences and REDUCE your productivity - something no one should be trying to do.

Servers should be on 100% of the time if possible. It's not a good idea to turn on and off server computers, which sometimes are designed to stay on even in time of hardware replacement.

However, if for some compelling reasons that you want to keep it down during non-business hour, such as to save money on electricity, you can automatically shut it down by simply writing a batch program with one line as below and schedule it to run at any time you wish.

shutdown -f

Also buy a $10-$20 digital on/off swich from Home Depot, put them on your computer's power supply to shut off power completely on a schedule. Most server power supplies still draw power when the computer is already shut down.

Just one note, don't be surprise to see one day the server would not turn on anymore.