Will remote desktop still work when Windows has turned off the hard drive? Can I still login remotely?
Yes, the computer should spin up the disk and allow you to remote in.
However, if the computer is suspended or hibernated, it will not.
The most important thing to do is test. Many computers, hardware, and OSs have different definitions of sleep/suspend/hibernate/etc. Be sure to test out how they work on your PC to see if certain options do/dont work for you.
Another thing to look into is wake-on-lan functionality (WoL).
This doesn't matter much – turning off the hard drive only causes it to spin down, but does not stop the OS or programs, and you still can log in. (The drive will spin up automatically whenever disk access happens.)
When the entire computer is in "sleep" or "suspend" mode, almost all of the hardware is turned off – the only part still running is the RAM chips. Since the CPU is turned off too, Windows is not running – you will not be able to log in remotely.
Usually, though, you can wake up the computer over the network by using Wake on LAN: when this feature is enabled, the network card is powered just enough to watch for the WoL magic packet. (WoL is designed for use within local networks, but it's possible, although somewhat complicated, to use it over Internet too.)