Not sure it would have had too much impact, or that it would have been a good thing if it had. Plotlines that are tied too strongly to real-world events can come off very klunky, espeically if they don't manage to capture real-world subtleties and complexities - see BSG's Caprica arc for an example of that.
DS9 might have used Iraq as an inspiration for post-Dominion-War Cardassia, but of course the analogy wouldn't be direct since the situation was not tied to any particular real-world event. That's the way I like these sorts of analogies to be used, so that would be a good thing.
They could have introduced a sectarian war on Bajor, which would have made the Bajorans more interesting so I'm actually in favor of that. Or they could have made more than they did out of the notion that the Dominion is the "bad" face of religion and the Bajorans are the "good" face, and set them directly in conflict with each other - something I wanted to see, and was disappointed never happened.
If you want the Federation to be the Bush Administration, we could have the Feds making atrocious mistakes and having to clean them up. Ironically, this is what actually happened in DS9 - take a good, hard look at the plot arc, and you could make a good case for the Feds actually starting the war by ignoring the Dominion's territorial integrity and CLEAR warnings to stay out. The Feds act like they own the whole damn galaxy.

No wonder they're always fighting their neighbors.
So maybe DS9 would have proceded as it did, but with more acknowledgement that the Feds contributed to the carnage, which is something we never got from DS9 except for a fig-leaf statement at the very end (the peace treaty ceremony).
To shoehorn the Iraq War into VOY's established planet-of-the-week format would have resulted in something gratingly simplistic - Janeway stumbles across a planet in the grip of sectarian war, etc, gives them a lecture and flies away. For this sort of horseshit to have some current events tie in would have been more maddening than usual.