Is it common or possible for fish oil to cause inflammation of the gums and/or mouth?

Not likely. Common side effects of fish oils are gastrointestinal like nausea. They also frequently cause a fishy taste. More significantly the can lead to excessive lowering of blood sugar in diabetics and bleeding problems in patients on anticoagulants, like coumadin, (warfarin) Aspirin or plavix. No oral/dental inflammation has been reported.
Possible alklergy? It does not typically cause gingival inflammation but the swelling can be a sign of allergy. I'd stop using it for a few days and see if the symptoms clear up.
Not likely. Fish oil supplements have anti-inflammatory properties so the supplement should not cause inflammation. If you are certain it is linked to the fish oil then you may be allergic to something in the supplement.