Is it possible to get business class airfare to Rome from the D.C area for under $2,000 tax included?

Certainly not within the next few months. June 15 to Sept. 15 is the airline high season when it is hard, if not impossible, to find discounts and deals coming from the airlines since planes during this time period tend to fly full. I did a few random searches for fall and early winter and DCA to FCO (Rome) business class was coming up around $3100 with taxes and fees. You could try airport code for Bush IAH and see if there is something better for the dates you are considering.

Not knowing your itinerary, a few tips. Try to fly out Tuesday through Thursday to not get hit with weekend fees. Try traveling after Sept. 15 but not within a week of the Thanksgiving, Christmas or new year's holidays. The first 2 weeks of November are a lull period and tend to have good deals. And if all else fails use your (or a friends) frequent flier miles for an upgrade into business class - still the best business class deal around.