I know exactly what you mean. When I was at college, every time I brewed a cup I got that oily film, yet, when I was at home brewing the same tea I would not get the film. For me, the issue was the water.
In my state, everyone has problems with "hard water," or water in which calcium and magnesium ions have dissolved. It's not a bad thing health wise, but it is obnoxious life wise. The water doesn't taste very good and everything is harder to wash, especially if the water isn't softened. Even with softened water, though, plenty of calcium and magnesium ions get through.
Here's the deal with hard water and tea/coffee making, though. When you heat hard water, the ions actually like it more and have a great time bouncing around in the aqueous suspension, knocking into other ions and forming small complexes. Once the water begins to cool, though, these complexes rapidly precipitate out (form small solids). This is what builds up on things used to heat water (tea kettles) and is often referred to as "scale." If the water is even fairly hard, though, you can also see the precipitates in the cooled water, often appearing as a white-looking "dust" or "dirt" in the bottom of a glass and as a thin, almost imperceptible film around the meniscus (surface).
When you brew black tea, tannins bind to this film and make it more easily seen--the oily film. It's still there in greens and whites, but a little more invisible. It won't hurt you at all, but it is disgusting.
Brita filters or other activated charcoal filters aren't exactly the most effective tool for dealing with hard water. Activated charcoal is primarily used to minimize chlorine ions and organic chemicals, which have different charges and properties than magnesium and calcium. This is why most household filters (including Brita) actually consist of two different filtration products. The problem is that the stuff that grabs calcium and magnesium has a shorter life, both because of the nature of the product and the fact that the water is GLUTTED with these ions. The filter turnover would make this type of product unfeasable in the long run.
You may want to look into getting a "reverse osmosis" system. The product mentioned in the previous post sounds a bit like one, but I didn't check that. Such a system will often have an activated charcoal filter, but will also have an additional semipermeable membrane. Basically, water is forced through the membrane, which acts as a very fine filter that only allows H2O molecules through. It's very, very pure stuff.
At my school, the water was not softened in the student houses, so we did the best we could with Brita. My parent's, however, invested in an RO system when I was 7. Their system was this giant thing that took up most of the space below the kitchen sink, and it had a separate spigot on the basin. However, this thing also allowed them to run a pipe from the RO system to the fridge, which had an automatic water and ice dispenser. That way, all the water making our ice and the stuff we primarily drank was clean and very convenient. I think it probably cost them $300-$400 initially, and the filters were probably $50 or so. I do know, however, that they only had to replace the membrane once every five years. I think they had once figured out that it would cost them less than a nickel a gallon in the long run.
Sorry for the behemoth of a post!