The oil must be removed (after machining) before further processing. However, delay cleaning if there is a delay or shipping involved, to avoid rust.
Re welding: “The most common cause of porosity is the presence of organic materials…Oil, grease and die lubricants should be removed by degreasing and washing operations” and the surface completely dry before welding.
—The Procedure Handbook of Arc Welding, 14th Edn., p. 6.3-15. However, some welders will ‘smoke’ through oiled material on non-code work.
Re Blasting: Oil makes a good binding agent for sand, etc., and would clog up any blasting equipment that recycles abrasive. At least, with properly pickled and oiled steel, there shouldn’t be any mill scale or rust to remove.
Re painting: Absolutely essential to degrease and clean prior to priming/painting.
And, if plating: The plater will clean as part of surface preparation; it is required by plating specifications. E.g., some require a ‘water break-free surface,’ meaning uniform wetting (no beading) when DI water is applied.