I could not get into that site either. It asked for login and password.
Absolutely NO NEED to discharge a monitor just to either take out, put in, or swap it. You only need to discharge it if you're going to work on it like pulling out the main board to recap it or replace a component.
Be aware though that if you're going to remove a monitor and that monitor was _just_ turned off and there is no ground strap/wire connected to the frame then there's a good possibility of a static charge on the front glass. So if you tip the monitor face towards you you may get a static zap just like you'd get if walking across a carpet and touching a doorknob. This sudden (and maybe unexpected) static zap may startle you enough to make you drop or jerk the monitor causing damage to you, the monitor, or both. For safety be sure the monitor has been off/unplugged at least ten minutes and that there was a ground strap connected to the frame.
Some newer monitors do have circuitry that disharge the high voltage right after power off, but remember that as a safety precaution (and to stay alive!) assume that the monitor is fully charged.