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.

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.

Nope: In fact it does the exact opposite.

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.