No.
Double click the .dmg file, install the app, and delete the dmg if you like.
Personally, I keep all mine handy, especially apps I payed for. If I ever need to re-install, I have them all on disk, and don't have to go searching. I recommend you do the same, it comes in handy sometimes. I've got backups of all my PC programs, and OS X programs, including widgets and preference panes. At least the ones I downloaded.