I'm flying into Gatwick and my connection to Shannon is through Heathrow.
Will my luggage be transferred by the airlines as SNN is my final destination?
No. Your luggage will not be transferred as you are connecting on the ground between two different airports. If you are travelling on a single ticket, you will still be protected and able to rebook in case of any inbound delays.
British Airways - by far the largest carrier serving both airports - for example has the following to say:
You must collect all your bags before leaving London Gatwick.
and, in case you've not made that journey before:
London's Gatwick and Heathrow airports are located approximately 30 miles apart. The recommended minimum connecting time between Gatwick and Heathrow is 3 hours.
The best way to travel direct between Gatwick and Heathrow is to take a National Express coach. It is fast, frequent, safe, reliable and great value for money.
It's quite hard to find a definitive source for any particular airline or airline pair (I assume the latter is more accurate in your case, as you'll be flying out to Shannon with Aer Lingus, and unless you're doing a very strange routing, you're not flying into Gatwick with them!). However, I think this is the kind of case where if this service was provided by an airline or alliance, they'd be shouting it from the rooftops as a differentiator against BA.