Sorry for the double post, but I decided to take the plunge and use Dashboard X to put the weather widget on my home screen and wanted to post my findings for anyone who cares. As far as I can tell, the widget refreshes every time you unlock your phone. It may also refresh every time you return to the springboard from an app, but I am not sure. The weather information doesn't change very often. In any case, over the day I used it, there hasn't seemed to be a noticeable hit on battery life.
All that being said, I ended up uninstalling Dashboard X for a number of reasons. The widget was slightly off center (even when using the auto-center feature), and often would move from where I put it on the home screen (especially after a respiring). It would cause an annoying jittery graphical glitch when unlocking my phone if I had scaled the widget to a smaller size (it was ok if I left it at full size though). The widget did not move down properly when the double hight status bar appeared during a call. Finally, the remove widget x button did not work most of the time, and I had to remove the widget with a respiring, which was very annoying. At this point, the tweak is simply too buggy for me to use. It's a shame, as I usually expect better of a paid tweak. Hopefully it will get better in the future.