I just realized that I have timecode breaks all in this video that I am supposed to edit for a friend. My question is: Is there a way to capture all of the video off of the tape without recording it onto another tape so I will have one continuous timecode?

nope. either make a firewire dub, or do it with capture now (no device control) and accept the bad tc.

No. If you want continuous timecode on the DV tape, you have to dub it.

Running FCP4 and above: FCP4 should be able to capture across timecode breaks without problems (i.e. audio sync). You might run into glitches if you want to re-capture off the DV tape, which may or may not be a problem for you.

could I capture the whole tape into FCP HD and then export back to tape and get a continuous timecode?

yeah you could do that. not sure why you'd want to that instead of a dub though. unless you just can't get your hands on another dv device to do the dub.