I tried installing Greenshot on a second machine to recreate my previous bug. The second machine has the same version of Windows installed on it - Windows 7, 64-bit. However on my second machine, I intentionally have my normal account as non-administrator. The installer doesn't try to get elevated administrator permissions and so fails in the last step of the installation process when it tries to write to the Program Files directory.
I can work around this, by right-clicking and saying run as administrator, but I think the installer itself should be requesting the Admin access.