If your email router does not process any emails but your inbox from Email Provider works just fine!?! Check event logs where Email Router is installed. You might see many repeated errors similar to mentioned error here below:
#27938 – An error occurred while checking for email messages to process in mailbox email@example.com. Microsoft.Crm.Tools.Email.Providers.EmailException: Error: The message exceeds the maximum supported size.
at Microsoft.Crm.Tools.Email.Providers.ExchangeWSConnector.LogEwsResponseErrorWarning(String message, ResponseClassType responseClassType)
at Microsoft.Crm.Tools.Email.Providers.ExchangeWSConnector.RetrieveMessage(ArrayList ids, Int32 index, ServiceLogger serviceLogger)
Check your inbox for the emails with attachments. Find emails with more than expected attachments size per email, take backup (if required) and delete them. Your Email Router should work just fine once this is done!
In my case an email with 100+ MB attachment size was creating an issue.