What would cause my filtered water to have a thin transparent film suspended in it?
Are you on a city water system, or a well?  Do you get the film from your tap?  If not, then it's something in the water line to the fridge, or in the filter.  In my whirlpool fridge, the "reservoir" for the chilled water is simply a coiled up roll of the plastic tubing.  Clean out the fridge, and take a look at that to see if you can see any obvious discoloration or contamination - it should be clear.  You could try replacing the tubing if you're suspicious.

If there nothing else obvious, then I'd try changing the filter, and run water through the new filter for a bit to clean everything out of the lines.  The filter, after all, is designed to collect gunk from the water, and if it's been a while, it could simply be full.