CI3: Apache vs Lightspeed Server

The only reliable answer would be to setup a test server running LiteSpeed with your app and run some load tests on it vs. a test server running Apache with your app. If it looks like you may see some improvement, the best thing to do would be to setup a production server running LiteSpeed and put both your existing production server and the LiteSpeed server behind a load balancer, if that's even an option for you. Then you could compare real-world results over time, and replace the Apache server later if it turns out that LiteSpeed really is the better option for your application.

Somewhere along the way, though, you should run a profiler to see if you can identify anything in your code that is slowing you down. The site/apps I've been running in CI for the last 4 years have never had anything improve performance as much as optimizing a handful of SQL statements, finding the occasional page which can gain significant performance improvements from optimizations which would be wasted elsewhere, or rethinking the way a particularly slow response is handled. The common thread in every case, though, is that I had to test and measure performance before I could determine the best way to address any performance problems I encountered.

Bonfire

Practical CodeIgniter 3
CodeIgniter Testing Guide