First, let's talk about how the main menu works. It is not 100% Lua based, but rather, it is a combination of HTML and JavaScript (JS). Lua is only called from JS context from the HTML.

To edit the HTML, you will need a text editor. You can use Notepad++[notepad-plus-plus.org], Sublime Text[www.sublimetext.com] or Eclipse[www.eclipse.org] or any other similar program to edit the HTML. If you don't have any of the above, Notepad will still work. For the purposes of this guide, I'll be using Notepad++, though the steps will be more or less similar with other text editors.

In your Games Library, right click on Garry's Mod, then click on 'Properties'.
Navigate to the 'Local Files' tab, then hit the 'Browse Local Files...' button.
You will be in the GarrysMod folder.
Navigate to garrysmod -> html -> template. This folder contains the HTML you need to edit.

Open main.html using one of the text editors mentioned. Don't open the HTML using your browser - it'll just show a broken webpage.

The file should look something like this. There is a lot of text in the file, but you don't need to interact with the whole thing.

At this point, choose whether you want to either hide the news feed, or remove the news feed altogether. Hiding the news feed allows you to easily restore it back, should you want to. Removing the news feed is much easier than hiding the news feed, and much more permanent too.

First, navigate to line 34 of the file. You will see the following text:
Replace "<" with "<!--" in the line so that it looks like this:
This will cause part of the HTML document to be "commented out", and may be colored differently depending on the text editor that you used.

After that, navigate to line 44 of the file. You will see the following text:
Replace ">" with "-->" in the line so that it looks like this:
This specifies the end of the "commenting out" we are doing, which will cause the </div> at line 46 to be colored differently, depending on the text editor that you used.
After that, you're done. Save and close the file.

This option gives you the advantage of easily unhiding the news feed. To unhide the news feed, navigate to line 34 of the file. You will see the following text:
Replace "<!--" with "<" in the line so that it looks like this:
This will cause part of the HTML document to be colored back to its original color, depending on the text editor that you used. One exception is </div--> at line 44 of the file, which will show the error color.

After that, navigate to line 44 of the file. You will see the following text:
Replace "-->" with ">" in the line so that it looks like this:
This will cause that portion of the HTML document to be colored back to its original color, depending on the text editor that you used.

After that, you're done. Save and close the file.

Simply delete lines 33 to 45 of the file, then save and close the file.

Launch Garry's Mod normally. If you don't see the news feed at the top right of your screen, it worked!

If you want the news feed back for whatever reason, just replace the edited main.html with the one you backed up. It should start to show again.

This is the first time I've ever written a guide. Hope it helps!