Is it safe to remove the "Install Xcode" application after installation?

Yes, it’s safe to remove the XCode installer, once you’ve installed the XCode tools. XCode tools can be downloaded for free from Apple’s website. If you are registered at developer.apple.com, you may also download the iPhone/iPad SDK from developer.apple.com, to be able to create iPhone/iPad software (the XCode tools that ship with MacOS X will only allow you to build applications for Mac OS).

According to this tread at the Apple discussion forum I can safely (re)move the installer application and still download it again from the App Store without paying again. It is unclear whether I will get update notifications.

This comment on a story about an attempt to upgrade Xcode with low disk space indicates that the move to an external disk may be tracked by the OS, and an upgrade via the Mac App Store can result in the moved application (even if it's on an external disk) being upgraded.

I did not get an upgrade notification for Xcode 4.0.1, even though my original installer application was still in the /Applications/ folder. I moved the installer application to an external disk (via sudo mv, because dragging it in Finder gave me the message "You may need to enter the name and password for an administrator on this computer to change the item named “Install Xcode”."), I'll report back what happens when I try to upgrade it via the Mac App Store.