No, it wouldn't.
A burst, cone, cylinder, or emanation spell affects only an area, creatures, or objects to which it has line of effect from its origin (a spherical burst’s center point, a cone-shaped burst’s starting point, a cylinder’s circle, or an emanation’s point of origin).
An otherwise solid barrier with a hole of at least 1 square foot through it does not block a spell’s line of effect. Such an opening means that the 5-foot length of wall containing the hole is no longer considered a barrier for purposes of a spell’s line of effect.
A doggy door causes the 5-foot length of wall (and door) to not be considered a barrier for the purpose of a spell's line of effect. It doesn't emanate 'through the doggy door'; it emanates through the 5-foot length of wall (and door) containing the doggy door. Their ears are silenced just fine.
-Hyp.