I must admit I have never turned timber having a mortice in it, but, I have turned hundreds of pieces of timber having voids, bits missing, splits, shakes, gaps, holes etc. I suppose these are similar although not as neat as a mortice.
Most of these pieces were mounted on some form of face plate, rarely between centres as I would guess you would turn something with a mortice in it, a chair or table leg for example.
Are you going to turn the area where the mortice has been cut? If you are, use a reasonably high speed on the lathe and more than likely you won't even realise it's there whilst you're turning. A word of warning though, if you are going to sand the piece on the lathe take extra care when working near the area with the mortice.
You should have no problem turning pieces with small mortices in, but you might get some vibration if the mortice is big and offcentre as in a farmhouse table leg for instance. In this case I cut a piece of scrap to fit in the mortice than hold it in place with masking tape while you turn it. has never failed yet!