A button to send a “your turn” reminder to the current player would be useful. I have friends who forget to go, and a way for us to fire off an email on the game page would be great.
I’ve had this thought before as well, but my main concern here is creating a spam machine. 
I think it’d have to be limited to once every X hours. Maybe a 3 strikes rule - a user can be reminded 3 times, and then anyone can kick them.
Both ideas sound reasonable to me