You should study the Reddit license, here:
https://github.com/reddit/reddit/blob/master/LICENSE
As a rule with Open Source software, if you obtain the Reddit software under an Open Source license, you must maintain that licensing for parties to whom you distribute your modified version.
My very quick scan of the Reddit license suggests that you don't have to make your source code available to your customers who access your website over the Internet, since you're not actually distributing software. (I won't swear to that point; someone will have to spend more time reading the license.)
There's nothing that prevents you from making a profit.
I'll also point you to a "list of links" I've compiled while preparing a presentation on Open Source:
"An Open Source Introduction: In Your Products, In Your Office"
http://yourlifeyourtech.blogspot.com/2014/01/open-source-references.html
You'll find links to the Free Software Foundation, Project GNU, and the Open Source Initiative on that page.