In light of the comments received on this answer (and the unnecessary downvotes) I've amended my answer as follows:
It seems Rails has now created a way to handle fonts in the assets folder. It's called font-path and works like this: 
If you add a custom font to your /assets/fonts folder, you can use the
  font-path helper to access the files within. This can then be used
  in your stylesheets & other assets using the font-path helper:
|-app/
|---assets/
|-----fonts/
|-----images/
|-----javascripts/
|-----stylesheets/
@font-face {
  font-family:'icofonts';
  src:font-url('icofonts.eot');
  src:font-url('icofonts.eot?#iefix') format('embedded-opentype'),
  ...
} 
This works for precompiled assets (which Heroku does anyway), and static assets. If you want the pre-Rails 4 way of achieving this, please refer to my answer below:
We've got fonts working on Heroku here: http://firststop.herokuapp.com (it's in demo still)
Here is our CSS for it:
#assets/application.css
/*-- Akagi Font --*/
@font-face {
    font-family: 'akagi';
    src: url('fonts/akagi-th-webfont.eot'),
    src: url('fonts/akagi-th-webfont.eot?#iefix') format('embedded-opentype'),
         url('fonts/akagi-th-webfont.woff') format('woff'),
         url('fonts/akagi-th-webfont.ttf') format('truetype'),
         url('fonts/akagi-th-webfont.svg#akagithin') format('svg');
    font-weight: 300;
    font-style: normal;
}
@font-face {
    font-family: 'akagi';
    src: url('fonts/akagi-bk-webfont.eot');
    src: url('fonts/akagi-bk-webfont.eot?#iefix') format('embedded-opentype'),
         url('fonts/akagi-bk-webfont.woff') format('woff'),
         url('fonts/akagi-bk-webfont.ttf') format('truetype'),
         url('fonts/akagi-bk-webfont.svg#akagibook') format('svg');
    font-weight: 400;
    font-style: normal;
}
@font-face {
    font-family: 'akagi';
    src: url('fonts/akagi-sb-webfont.eot');
    src: url('fonts/akagi-sb-webfont.eot?#iefix') format('embedded-opentype'),
         url('fonts/akagi-sb-webfont.woff') format('woff'),
         url('fonts/akagi-sb-webfont.ttf') format('truetype'),
         url('fonts/akagi-sb-webfont.svg#akagisemibold') format('svg');
    font-weight: 500;
    font-style: normal;
}
We put our fonts into the /stylesheets/fonts folder
We just do the standard precompile fonts stuff to get all CSS working on Heroku, and it works. I suspect your paths are not correct. Maybe try moving your fonts directory into your /assets/stylesheets folder :)