I'm using a library in my program and that library has a MIT licence. I don't download the library itself I just use it via a CDN provider. Do I need to include the library's licence? If yes, how should I do it, where should I put it? (I'm using GitHub for this open source project.) Do I need to include it to the downloads (Releases page on Github)?
The only legal requirement you have is that you don't remove the license text and copyright statements from the files you download from the CDN.
As a matter of goodwill, it is good to mention in your project documentation, for example in a readme file, that you are using this library, which license it uses and where the source code is hosted. There, you don't need to include the actual text of the license, but only the (common) name under which it is known.