If your site is database-driven I would be very surprised if your bottleneck would be the application framework. "Fast" as in faster development is what I would worry about rather than "fast" as in speedy handling of requests. Significant optimization is better done by caching strategies and optimizing your database access.
Besides database access your own code will be where most of the time for each request is spent (and even that is usually not significant compared to database access), the framework will likely not be affecting the time spent on a request, unless it is really badly written.
It way be better to look for a framework which has good caching support (which Code Igniter may have, I don't know), that will almost always save you more time than the few milliseconds you could shave off the request handling by using a slightly faster framework.
Have a look at the Zend Framework too, it has the benefit of being PHP 5, whereas Code Igniter is still PHP 4, as I understand it. That may be an issue when it comes to speed, but in favor of which framework I don't know. Zend has good caching support and a database profiler that can help you find where your bottlenecks are.