Do you need a search button next to a search box on a mobile website? On the iPhone for example the Enter button appears next to the standard ios5 ui keyboard.
Are you providing suggestions as the user types?
If there is immediate feedback as the user types then no, they can select either the suggestion or their typed search term.
If the feedback is non-existent, a button is helpful and intuitive more so than the enter button. You mention the iPhone as an example however if this is intended to be a mobile site, that mobile device can be any number of devices. If you were deploying this in to a given app store, marketplace, etc... the approach could be different.
All Android and iPhone devices have the Go/Enter feature on the standard keayboard. However, as Aaron McIver mentioned a different mobile device could enter the website and there is no guarantee that it has that feature in the keyboard.
So I would definitely recommend adding a button for Search if you want the site to be cross-platform compatible.