Not really. Here's why:
Pictures taken by the iPhone are saved in the original resolution.
Pictures that you synch from your computer to display on your iPhone are NOT the original photos, nor are they of any resolution that's practical for use. Basically, when you synch your photos to your phone, you're syching tiny little "thumbnails" of the photos, not the photos themselves. Thats why you can synch hundreds, if not thousands of photos on to your phone. There'd be no way to do that if they were of the original resolution. The "thumbnails" on your phone are of no use other than to view on the phone. They are not of sufficient resolution to allow for printing or anything else.
What you need to do is take the original photo files from your original computer and move them to a thumbdrive or portable hard drive and then transfer them to your new computer.
Hope this helps.
Once you synch your phone to a new computer, don't check synch photos, and your photos will be removed. You may receive a message telling you that those photos will be deleted.