If the tool is not sharp, and mounted on centre height, it will generate excess heat, which will result in smoke.
I part off under power, with an inverted Zero top rake parting tool. Applying ordinary lubricating oil by brush, or oil can, does not produce smoke. But the speed is low and the feed is 0.060 mm/rev.
Are you running at too high a surface speed for the material being cut?
Don’t try to use industrial surface speeds / feeds on a hobby machine, which is not designed for those sorts of cuts or feed rates, and lacks the power.
A mini lathe is likely to have a 500 watts brushless motor, not a three phase 20 hp motor.
Howard