No way Gore invades Iraq. Yes, presidents have different incentives than private citizens, but those incentives almost all run towards caution, not military aggression. It's perfectly reasonable for me, a private citizen, to idly talk about how we should invade Country X (see various newspaper columnists, pundits, etc., since the dawn of media). It's much more serious for the commander-in-chief to do so, which is why they tend to be more cautious about such things. The "President Gore would be different from Citizen Gore" in this case has always been a ridiculous argument.
Nor would Lieberman make a difference in this respect; he's the VP, and Gore is not Bush when it comes to interest and ability in governing and foreign policy. The VP has only as much authority as the President gives him, and Gore (who chose Lieberman primarily for electoral reasons) is not going to give him authority over foreign policy. Nor is it clear that Lieberman would be making the same arguments as VP in a situation where a war with Iraq seems unlikely as he did as a Senator in a situation where war with Iraq was being heavily promoted. If war with Iraq is never on the table, he seems unlikely to be advocating for it.
A Democratic administration probably satisfies itself with a heavier military involvement in Afghanistan, including greater numbers of boots on the ground from the beginning. Not only is there no distraction with Iraq, but a different Secretary of Defense probably won't share Rumsfeld's "do more with less" vision of military doctrine, and will thus want to put more troops into securing the situation from the beginning.
An interesting question will be how the Republicans react. On the one hand, the rally-round-the-flag effect will still be strong. On the other hand, the Republicans would be quite justified in pointing out that the Democrats had been in power since 1992, and had failed to prevent the attacks, unlike OTL where Bush could plausibly claim not to be responsible after 9 months on the job. Would they support the administration, criticize its unpreparedness, or try to out-hawk it (probably not with calls for an Iraq War in particular, but with more calls for stricter security measures/"enhanced interrogations" and a more hawkish approach to the Middle East in general)?