There are multiple reasons for this kind of issue. Typically it is either has to do with the way Heroku precompiles the assets or that the CSS is not pointing to the proper file if the asset is stored on S3.
One such being this:
http://natashatherobot.com/rails-4-heroku-assets-not-found-css-image/
Another being this:
Can't get CSS working on Heroku using Rails 4 with bootstrap-sass gem
Edit
Okay. It looks like you need to change the way you are managing your assets.
One solution is to use S3. A good explanation on how to do that: https://devcenter.heroku.com/articles/s3
I haven't used this gem but it looks promising:
https://github.com/rumblelabs/asset_sync
Another solution is to keep your local asset file in synch with production. You could do this by pulling down a copy of your asset directory from heroku each time you do development.