Limit the number of registrations by field I am using Entityforms to take registrations for events. There is limited space and the form allows users to select which event they would prefer to attend.

Is there any way (rule, etc.) to disable or remove certain dates once the limit has been reached?
You can disable specific dates in date popup by implementing some custom code as per shown in jQuery UI DatePicker: Disable Specified Days. To work with jQuery in Drupal you can check the date popup alter example here.

OR you can try Work Calendar Module with Date Restrictions Module (both are in dev and not suggested to use on Production unless working without any error).