For the Current version of QuickTime:

Right-click the icon and select QuickTime Preferences
Next Step is, select Advanced tab.
Locate the option Install QuickTime icon in system tray in the “Tray Icon” area near the bottom and uncheck the box next to it.
Close the preferences Window by selecting OK
This should remove the icon from your tray and it should not be there the next time you restart your computer.

For those of you who use Older Version of QuickTime

Right-click on the icon and select QuickTime Preferences
Next go to, select Browser Plug-in
Then Locate the option QuickTime system tray icon and uncheck the box next to it.
Now Exit the options by closing the window (click the X at the top right, there is no OK or Cancel

This will remove the icon from your tray and it should not be there the next time you restart your computer.