Is it possible to visit Dome of the Rock for non-muslim tourists these days?
Last time I tried was in 2005, and back then it wasn't.
According to Wikipedia, the Dome of the Rock complex (Temple Mount) was reopened to non-Muslim visitors for limited time and days of the week (for example, non-Muslims are not allowed in on Fridays).
Frommer's has a detailed description on how to get tourist access to the site. You may be able to get into the complex and the museum, but probably not into the actual dome.