I have this receiver, here's what will happen. First, I'm 99% certain that your plasma is 720p, not 1080i. AFAIK, all plasma, LCD and DLP TV's are progressive scan, not interlaced. 1080p is pretty recent, so that's why I believe you have a 720p plasma. This is important, depending on what source devices you'll be using.
The HDMI to DVI cable should work just fine. Any sources that you plug into the 605 by HDMI will have the video passed to the TV straight through, no alterations made by the Onkyo. For example, If you have a Blu-Ray or HD DVD player connected to the 605 by HDMI, and the player is set to output 1080i, then the 1080i signal will pass through the 605 to your TV, which will scale it to 720p.
Now, if you have something connected by component, then the 605 will intervene and do some video processing before sending to the TV over the HDMI->DVI cable. If you had your Dish box connected to the 605 by component cable, and the output of the dish box set to 1080i, the 605 would convert that signal to 720p before sending it out over the HDMI out port.
So, seeing as you have a 720p TV, either case shouldn't result in any potential signal loss.
Also regarding your TV, I wouldn't be in such a hurry to replace it. Don't get caught up in all the 1080p hype. There are lots of factors that go into making a nice picture, and resolution is probably the 3th most important (after contrast level, color saturation and color accuracy). I'd take a 720p Pioneer Kuro plasma over any 1080p LCD panel available, even though its lower resolution.