Question: I am building a site that will (obvisouly) have a front end public portion that I want to drive with Concrete5, but then it will also have a members section that I would like to build with Code Igniter.

Does anyone have any idea how I could do that?

I guess I could just throw the Code Igniter code into a sub directory, but would I run into any issues with that?

Answer: 
i can't see why not. 

As you suggested a separate folder on the site would be one solution.
(you might have to tweak the .htaccess file (if you are using one) to ignore the other cms/framework

Another solution would be to have separate subdomains,
eg example.com and members.example.com
I ended up using a sub directory for Code Igniter. I had to change the .htaccess file and Mod\_Rewrite rules for apache in order for the Code Igniter "pretty" URLs to work correctly.

I will post the code later if I remember.