Checking bags to final Destination

Yes, if the journey was booked as one ticket.

Our trip is booked on separate tickets-one for Aer Lingus and one for British Airways. Does this mean that I must collect baggage in Dublin and check it in to British Airways? Thanks for your response!

afraid so

I would still ask at the airport when you check in, just in case. Airline interline agreements between the major carriers are meant to facilitate this kind of transfer, even when you buy tickets separately. For instance, I had a heck of a time convincing a Greek island airport agent that Delta had an interline agreement with Olympic Airways, but once I was able to provide the code-shared KLM flight number for my Delta flight, then my bag was checked all the way through (Kos-Athens-Amsterdam-Seattle).

According to Aer Lingus' site, they do have an interline agreement with BA, but specifically states that your journey must be on the same ticket for them to transfer you bags:
http://tinyurl.com/qfrlnkz

But I would call Aer Lingus and ask them about your situation, I thinks there is a good chance they can do anyways:

1 (516) 622-4022

So to be clear...you booked two - or perhaps three - separate airline tickets?

Then no, you can't check them through, probably, unless as previously stated they have some sort of arrangement with each other. You'll have to find out, and I'd get that info in writing.

Also, if your first flight is late enough to cause you to miss your connecting flight, there is no obligation for Aer Lingus to get you to Vienna. You'll need to buy a new ticket. Ouch. Now if you bought one ticket on two airlines (aka, one itinerary) as codeshare partners but you have three flights, then yes - they will (probably) check your luggage through and get you to Vienna if they cause you to miss a connection.