If I fly on Aer Lingus from Shannon to Heathrow, and then catch a connecting British Airway flight to Paris, will I be able to check my bag all the way through without clearing customs at Heathrow...? I am wondering how much time I'll need between flights at Heathrow... Can I do it in an hour and a half?
if this is all on one ticket your bags will be checked through to Paris
you will not encounter immigration controls at LHR on flights from Ireland
if BA or Aer Lingus are selling you a ticket with a 90 minute connection then that's OK
You will be changing terminals. if you go to Heathrow Airport Flight Connections and plug in your flight information it will give you step by step instructions on what to do at Heathrow and the minimum suggested transfer time.
Is this after visiting Ireland or after landing in Shannon off a flight from North America? If you're traveling around Ireland first, you can get direct flights to Paris from either Cork airport or Dublin airport. If this is the continuation of another flight, then it sounds like a long day.
I'd pay extra to avoid Heathrow, a huge and arduous place. But flying non-stop out of Dublin airport should be at least 50 per cent cheaper than Shannon anyhow.