I have barely played any of the DLC in Mass Effect 2. If I do not get around to playing it (and thus no save file containing data about the DLC events), will I miss anything in Mass Effect 3 (such as character encounters, items, missions, etc)?
Yes there is.
If you don't have the DLC for Kasumi you will miss out on her interaction for the hanar mission:
Zaeed, will also make an appearance in the game on a quest involving a Hanar and Cerberus, resulting in the volus bombing squadron as a war asset.
Other smaller dialog changes include conversations with Liara (shadow broker related), and the outcome of the Arrival DLC.
If you saved Gavin Archer's brother, David, in the Overlord DLC and let him stay at Project Overlord, you'll run into Gavin again in ME3 on a side quest involving Jacob and he will be a war asset.
Other DLCs such as Bring down the Sky from ME1 will allow you to increase your war assets. In this case, if you let the Batarian leader live, you can recruit his fleet.