Assuming that your users are on windows PCs, change the following 
registry value:

In HKEY_LOCAL_MACHINE\SOFTWARE\ORL\WinVNC3,
"DisableTrayIcon"=dword:00000001

This "registry hack" only works on some flavors of VNC servers,
but not all: TightVNC is supported, RealVNC is not. See here for 
the whole list of Flavors of VNC and VNC Add-on's.

Thanks to Nick Palmer for the contribution!

An alternative and more drastic measure to prevent the VNC
icon from appearing in the system tray is to modify the
a registry entry specific to Windows itself. Making a mistake
in here could prevent Windows from ever working correctly again, 
so please beware.

If VNC is running as a service, you can remove its service helper
key from the Windows registry. That is, via regedit, remove the 
"WinVNC" value which can be found in one of these:

HKEY_LOCAL_MACHINE\SOFTWARE\Microsoft\Windows\CurrentVersion\Run
or
HKLM\SOFTWARE\Microsoft\Windows\CurrentVersion\Run
or
HKCU\SOFTWARE\Microsoft\Windows\CurrentVersion\Run

Again, this is very risky. But it should work.

The third method of "hiding" the VNC tray icon requires some 
developers tools: using a resource editor, modify the WinVNC 
icon so that it will show the same icon both before and after 
a client is connected.

Thanks to Michael Roland for the contribution!