I've read that Litespeed is a lot faster than Apache. I use mostly WordPress and Joomla sites and am looking for a new server. Will I see a significant increase in speed of WordPress being on a LiteSpeed server?
No you will not see a significant increase in speed just by being on a litespeed server. They claim it serves PHP 50% faster (than mod_php) but they haven't updated their SAPI in over a year. PHP-FPM will blow it out of the water any day.
Stick with open source and if you want a web server that really is faster take a look Nginx
There are a lot of LiteSpeed haters out there, and I can only speak from experience. I've found that LiteSpeed handles memory much better than Apache. I switched to LiteSpeed on a VPS that hosts about 20 WordPress websites and the reduction in server wait and page loading time was significant. Switching to LiteSpeed was also very simple --- it integrates well with WHM/cPanel, and in my case my hosting company took care of installing it. I'd highly recommend you give it a try.
Sorry if I go a little bit off topic here, but I just wanted to comment on this subject although maybe most of these things you will already know.
When you want to speed up a WordPress site there are a few steps you can take that will make a major difference.
Another thing that will make a major difference will be a CDN. You can use CDN's such as Amazon S3 and CloudFlare to distribute your static files to servers all across the world and serve them to users closest to those servers. This will increase the speed of your website dramatically.
If that is still not enough than you can resort to changing to a different web server. I can not tell you a lot about LiteSpeed, but I do have some experience with Nginx and I think they will perform almost the same.
From my experience Nginx is a little faster than Apache, I had the exact same WordPress site on two servers, and Nginx did a lot better than Apache. I read that it had something to do with the way Nginx spawns the processes, but that is a topic for ServerFault.
To conclude, changing your web server can make a difference, but be sure to have your site optimized before switching.
I use an Apache with PHP-FPM process manager for serving PHP, and Nginx for serving the static content, with a Varnish cache in front of the entire thing.
You see a lot of hosting companies offer PHP-FPM + Nginx like this nowadays. I don't have any benchmarks on how this compares with LiteSpeed, but I have used both setups, and they both work great. The important thing is I would make sure that you're not using the default mod_apache process manager if your using Apache. Other than that, it shouldn't really matter if you choose Apache or LiteSpeed.