Removing Upload Form

http://wordpress.org/support/topic/53893#post-294963

Or use the following plugin (which does much the same thing, only no source editing required):

Hide Post Upload*:
download file | view source

Download the file, upload it to your plugins directory, activate Hide Post Upload under plugins, then…no, that’s it.

* One of a neverending line of single-task-minded plugins from the folks at szubNet! :p