Created a new teamfortress.tv thread since I decided to continue to update this mod.
http://www.teamfortress.tv/35222/no-hats-mod

Hrbls recently stopped updating the No Hats mod so I took over to prevent it from dying out.
hrbls thread: http://www.teamfortress.tv/16179/no-hats-mod
Original thread: http://forums.steampowered.com/forums/showthread.php?t=2041907

Latest version can be found here:
https://github.com/jeebsAU/No-Hats-Mod
Latest Version: 16-May-2017: Removed the Rainy Day cosmetics and the Snack's Summery Ultiduo tournament medals.

Moved it to a GitHub repository. MediaFire is behaving weird and won't allow me to upload the newest file. All future updates will be pushed to GitHub from now on.

Unfortunately this version doesn't fix Pyro's having no heads, Scouts having no feet, etc. so if that's something that bothers you then I recommend you download Springrolls Headsfeet version which fixes this issue.

To install simply navigate to: Steam\steamapps\common\Team Fortress 2\tf\custom\ and place the .vpk file there. After that just boot up TF2 and you should no longer have hats showing.

Enjoy your hatless TF2!

Note: If you see any hats in game with this mod on, feel free to tell me so I know what else has to be removed.

I have now officially updated the Headsfeet mod. Download it by following the link below. It will download a .zip file. Open that .zip file and extract the .vpk file to:
Steam\steamapps\common\Team Fortress 2\tf\custom\
I'll do my best to try to keep this mod updated along side the No Hats mod.
Download link: https://github.com/jeebsAU/Headsfeet

Updated to remove Season 1 Pass Time medals
https://www.mediafire.com/?1bdky9ca9th5266

That is a known issue with this mod unfortunately. I'd recommend waiting for headsfeet to get updated if Pyro's without heads and Scouts without legs is something that annoys you.

Here's an updated version of No Hats for the Meet Your Match update which added the PASS time items and Disgaea items.
http://www.mediafire.com/download/77uakx8gp7dgucc/