Jim...that is too gross...and definately not normal. But a little more info might help.
And maybe others have experienced this.
Is the film present when the tea is still very hot, are they flavored teas, how do you prepare your tea, and do you experience this any other time, do you add anything to the tea other than water? Have you tried alternative water sources for your tea such as bottled water?
If I were you, I too would want to get to the bottom of this.