Does the Turkish getup create conditions for shoulder impingement the way wide-elbow benching or rings slipping away from you during dips do?
Poor soft tissue quality, poor mobility, and poor muscular control create conditions for shoulder impingement. There are exercises that place the shoulder in a position of greater chance for impingement, especially if you have one of more of the three previously listed conditions, but I don't see the turkish getup as one of them. If you have the mobility and muscular control required to perform the exercise correctly, there shouldn't be a problem.
Mostly no, unless you have terrible T-spine mobility. That'll be obvious when you stand up and your arm can't be vertical, with bicep in line with your ear. If you can see your hand when you stand upright, you're doing it wrong, and likely have shoulders that will find get ups aggravating. Solution - kettlebell armbars, brettzels, bridge, and do half get ups until it clears.