You do not need MS Office on your computer to run Limewire. I deleted every bit of it(Office)ages ago!
This thread in OSX support might help
http://www.gnutellaforums.com/showth...threadid=42872 Its a bit technical and might not be related to your problem.
Also check that what you downloaded is titled LimeWireOSX.dmg as there has been a case of it having a different title and being launched unsuccessfully in Excell now there's a strange coincidence!
If your downloaded file is correctly titled it should open without any tie up with quick time afaik. If you control/click on it it should say "open with DiscImageMounter (default)" If it does not let us know what it does say.
If you do not want to keep MS Office, it seems like agood idea to delete it and try again. I hope this helps Let us know if you need more help!