Here's a simple way to hide the YouTube live chat window and block it from loading altogether. This should also result in a small improvement to page load speed when visiting YouTube video pages that have live chat.

Get an Ad Blocker

To block the live chat, you'll need an ad blocker extension or plug-in for your browser. If you don't already have one, you can get one from your browser's extension/plug-in page. I recommend µBlock Origin for either Chrome or Firefox. Any one that supports the AdBlock Plus filter syntax will work.

Adding a Custom Filter

Once you have the ad blocker installed, you'll need to add two lines to the custom filter section. You can do this in the "My filters" section of uBlock, which can be found depending on your browser at:

Chrome
More Tools > Extensions > uBlock Origin Options > My filters

Firefox
Add-ons > Extensions > uBlock Origin Options > Show Dashboard > My filters

Add these two filters on their own lines (copy and paste them) and then hit "Apply changes":

https://www.youtube.com/live_chat
www.youtube.com###chat
You shouldn't need to restart your browser.

Reverting Back

If you want to revert back to allowing YouTube live chat, you'll have to remove the filters from your custom filters.