Does a permanent that was a creature until end of turn remain tapped after it stops being one?
Yes: Being tapped/untapped is a part of the status of a permanent and is independent of its type (creature/not creature) or other characteristic.

110.6. A permanent’s status is its physical state. There are four status categories, each of which has two possible values: tapped/untapped, flipped/unflipped, face up/face down, and phased in/phased out. Each permanent always has one of these values for each of these categories.
110.6a Status is not a characteristic, though it may affect a permanent’s characteristics.

In your example, Sarkhan, the Dragonspeaker will remain tapped even after your turn ends.