I've done it at LHR coming from RDU going to DUB. When you pass through the Flight Connection Centre at LHR, you will be routed to a separate queue at T1 where your passport will be examined and a photograph will be taken. Upon arrival in DUB your passport will be checked again, assuming you are not a citizen of the EU. The DUB check is brief, but LHR could easily take the entire 60 minutes. It can be a long walk to reach the Ireland boarding gates at T1. Nevertheless, 1:55 should be plenty. Flights between LHR and DUB are frequent enough that even if you miss your connection, you won't have to wait long for the next flight.

You have the option of passing through UKBA into UK landside at T3 and then reentering at T1, but I wouldn't recommend it.