Using images instead of text has been around as long as the <img> tag. Designers frustrated by the lack of typographic control routinely created designs in image editors and in worst case scenarios, created websites entirely from images. Today, this is almost universally considered bad practice, as the content of an image cannot be indexed by search engines (without horribly dense alt tags) and the bandwidth overhead is increased substantially, both with the extra markup required and the image sizes themselves.

Nevertheless, image replacement still remains in use, and is in fact, extremely common. Every website using a logo in their header is using a form of image replacement. The use of an image representing the brand coupled with an <alt> tag is ubiquitous on the web. An extended and superior approach is to include the text contained in the image as a <span> or <h1> in the HTML, and use CSS to hide the text but insert a background image.
There are a variety of methods to accomplish this. These approaches maintain accessibility and should not adversely impact SEO.

this is an image, not text.

The example above is inserted directly into the html, and has a alt tag. The example below uses CSS to insert a background image into a <h5> and wraps the actual text in <span class"hidden-text">.
With CSS the visibility of that class is set to "hidden", hiding the text, but leaving the background image visible.

The technique is simple and effective for use with static elements such as an identity, but can become labour intensive if, for example every heading was to be styled in this fashion. It would also mean that many website updates would also require image editing. FLIR (FaceLift Image Replacement) was developed by Cory Mawhorter to solve this problem by making the process dynamic. Similar in concept to sIFR, FLIR replaces text with png images created dynamically on the host server by a php script (GD or ImagMagick). The use of php could be viewed as a disadvantage, as servers running .Net cannot run FLIR.
Mawhorters site is no longer online, so it was not possible to test this technique. It is unclear if FLIR is still being supported, but a version of it has been forked to run as a WordPress plug-in.

In summary

Image replacement will probably remain the most effective method of branding a website. Its suitability for typographic use is questionable however, particularly when other more effective techniques exist, such as sIFR, Cufon and @font-face.
It may be justified in cases where the font EULA does not permit the use of those technologies.