Actually that's what they use to test engines for bird collissions... turkeys shot into the engine running full blast, quite a sight to see.
They typically use ducks for the bird ingestion test, and the ducks are alive. A typical method used to disable the duck is to spray it with a CO2 fire extinguisher. That will "stun" the duck but will keep it alive until its ingestion. Dead birds don't seem to accurately simulate the action of live birds being ingested into an engine, and that's why live but stunned birds are used.