Azure WebJob running old code
I had a similar problem. We deploy using a stage environment in Azure and it turned out that the "old" WebJobs (running code with an old version of the entity framework model) where still running on the queue. These jobs where then fetching messages and consuming them. To add to the problem the exception was consumed in a try catch and the status of the WebJob was success.
Check if you have a stage environment (add -stage to the Webapp name) and if so go in to the Azure management portal and stop them.
Note, it is not enough to stop the Webapp, you must stop the WebJobs directly. This is done (in the new portal) under Settings->WebJobs and then right-clicking on the webjobs name selecting stop.