Yes, it's true - that's why screen savers were invented.
How long it takes depends on a lot of things, but back in the days of DOS when people would leave a menu on their screen most of the day, I would often visit clients where the menu was readable with the machine turned off. Basically the phospors fade with use over time. If the image is constantly changing, the fading is distributed over most of the screen, but with a constant image, the brightest parts of the image will eventually get burned into the phosphor coating.