Primarily displacing and out-competing native wildfowl (and possibly some native rodents) that depend on the same foods.  There's also the issue of them destroying human property.  And, as others have pointed out, releasing domestic animals into the wild is a violation of both Anchorage Municipal ordinance and state law (5 AAC 92.029—“Permit for possessing live game”).  Under some conditions, such as when unrestrained domestic animals are observed harassing or harming wildlife, it's legal to shoot them (per AMO 17, if I recall correctly).

We're asking for it if we allow individuals to flout these laws--there are already to so-called 'animal welfare' groups petitioning State Fish & Game permitting office and the Anchorage Assembly to exempt them from these laws so they can maintain unrestrained outdoor colonies of feral cats--and both groups are already maintaining such colonies illegally.

Shoot the cats, then the turkeys.