The sites seem to work without any visible issues.  The Event Viewer on the server reports 404s on the ScriptResource.axd and WebResource.axd files.

This is an issue with machine keys in a load balanced environment.

The recommendation is to standardize the machine keys on all the servers in the load balanced configuration.

The instructions for modifying the keys can be found here .
 
The key is to apply the same IDs to all the servers in the configuration.