The fence gate has no opening or shutting sound when activated by pressure plates, levers, buttons, or any other redstone related items. This affects all variants of gates and yields the same issue when tested in combination with all variants of pressure plates. See photos attached.
Linked issues
relates to
Attachments
Comments

Still an issue as of 0.14.0 release.
Affects 0.14.2

Affects 0.15.0 build 1

Just a theory: Perhaps the same reason it doesn't play a sound is the same reason opening fence gates with redstone circuits will not trigger an Observer? (See discussion on MCPE-14710) It could be that the fence gates don't trigger a block update when powered by redstone, though I'm not sure if its sounds are tied to that or not.

Confirmed for 0.15.90.2. (Tested on Pocket Edition.)

Still affects 1.0.2
Affects 1.0.4.0 according to MCPE-20110.
Affects 1.0.4.1.
Affects 1.0.5.0.
Fixed in 1.1.0.0
Confirmed on Samsung Galaxy Note 5. Android 5.1.1 Version: 0.13.0 Build 5