No, they don’t.
They also don’t get a notification if you screenshot an Instagram Story nor a direct message. However they can know when you saw them in both cases as well as knowing who saw the video. The view counts if they click on the video to get audio or watch it for more than 3 seconds.
It’s the same with Facebook videos, posts and Stories and one of the main differentiation points between the Facebook empire and Snapchat’s model of operation.