How to remove stubborn iron stains from ceramic shower tiles?

For one thing, I'd have my water re-tested. If it looks like iron, and stains like iron, it's probably red-water iron in the water. There is a chemical called "Rust-Aid" which is sold at Home Depot, it's used for power washing driveways. I'm sure that if you apply some of this full-strength to the stains, they'll come off. Muriatic acid can be VERY dangerous to work with, the fumes are toxic and you can get a chemical burn if you get it on you - be extremely careful if you try using this. It's really nothing more than diluted hydrochloric acid and is used for concrete cleaning, etching, etc.