Hi Kinghob,
My best recommendation would be to change the mailboxes so that they don't have a $ in the name ... but if that won't work you should be able to put the ` mark in front of the $ sign to escape the $ ... thus powhershell will just see it as a normal character.
You will most likely have to export all targets with a $ in the identity to a CSV then insert the ` character in the correct location. Then you can user import-csv to get the new information in to a variable and pipe that to remove-mailboxpermissions.
Personally I would change the mailboxes so that they don't have the $ sign anymore ... otherwise you will just keep running into this issue.
Hope this Helps
Matthew Byrd