Depends. Windows XP and 2003 will deactivate itself when it sees the new hardware, but other Windows versions don't have activation so they'd be fine. The biggest problem will be that the system you put the drive into will have completely different hardware than the old system, so you'll need a completely new set of drivers, and things can get flaky if you have drivers and applications for multiple different video and sound cards installed at once. (if you go from nVidia to nVidia, no problems... same with ATI... but you don't want nView loading itself on a Radeon card!)
It's best in terms of getting your system running efficient and stable to reinstall windows, preferably formatting or at least deleting the old version and all temp files and application directories first. That way your drive isn't filled up with old crap that can pile up or get in the way, and you don't have any random stray DLL's from hardware or apps that aren't installed sitting in your windows system folder.