Erm, why? So you leave your computer running for weeks without turning it off? Bit of a waste of electricity....
As long as the monitor is off, it's not much electricity being wasted.
In fact default windows settings even when the computer is plugged in will first turn off the monitor, next turn off the hard drive, and then put the computer to sleep for you.
The main reason I can think of to keep Civ V running is to save time doing all the following:
1. Computer contacting steam
2. Civ V launching
3. Loading the game file.
The main disadvantage is that there appears to be a small memory leak in the section of code dealing with display, perhaps in DirectX itself. While for a few hours it doesn't matter, eventually (if left on for days) it can cause a crash.