Is there any way to deal with helicopter gunships other than shooting them out of the sky?

As far as ground vehicles go, there are lethal and non-lethal options to destroy and capture respectively.

Peace Walker had methods for dealing with helicopter gunships that would lead to capture, such as shooting the canopy open to expose the pilot, and then tranquillising them.

Does Phantom Pain also feature non-lethal options when it comes to dealing with helicopter gunships? Or is destroying them the only option?

Destroying them is the only option, or don't get noticed by them.

Destroying a helicopter will not count as a kill, nor add Demon Points, unless it crashes on somebody else, as the game considers the helicopter to be unmanned.