You never want to use grease on ways or other parts on a lathe that are exposed to chips and dust. There is special oil for the ways, slideway oil, which is relatively thin and non-sticky so it is easy to clean off. The oil has about the same feeling as oil for air tools.
In case an oil like a straight SAE 30 engine oil was used it would be difficult to remove fine dust like from cast iron. It is also good to cover the ways and put some paper behind the work piece in the chuck when turning cast iron.
Grease is only used for protected parts like e.g. closed gears and the backside of a chuck scroll wheel.