The meaning of "finishing room" depends on the industry. For example, in the fiberglass thermoset resin business (producers of auto body panels and other large fiberglass plastic parts that need a smooth surface on both sides of the panel), the manufacturer of the SMC (Sheet Molding Compound http://composite.about.com/library/glossary/s/bldef-s4841.htm) that is used by the panel/part producer uses a finishing room to "age" the product before shipping to the panel producer. Check Wikipedia http://en.wikipedia.org/wiki/Sheet_moulding_compound for an extremely short overview of the SMC process.
 
Also, other process industries, like wafer and computer chip manufacturers, could have a finishing room (though I have no direct knowledge of that).