I removed two officers earlier in the year which was quite easy since they hadn't paid dues and didn't even attend the first meeting (which meant they weren't technically officers to begin with). Then, the other officer I removed had paid dues but hadn't shown to two meetings or communicated via phone, letter, or e-mail (same with the previous mentioned two) so I wrote a formal letter stating if I didn't hear from her by such-and-such date, I would consider it her resignation. I did not hear by the date and sent a follow-up letter confirming her resignation and removal from office. This meant no formal procedure with higher district and state offices. I did keep the principal informed though. Hope this helps.
If by some crazy chance there are no by-laws then Roberts Rules of Order (do a google on it) has a website that can give formalities on this issue.
Otherwise, you can ask for the officer to resign as long as you have majority and warrent cause. Our by-laws ask that there be a hearing to discuss the matter and then a formal request is made.
If your group has by-laws, your Secretary should have a copy of them. As President, I have a copy with me at all meetings in case a question would come up.
Our by-laws really don't have a procedure for impeachment. They do have a portion covering duties of officers and expectations for attendance that could allow us to remove an officer under certain circumstances.
If this desire to remove an officer is shared by all members of the board, I would think that they could ask for the officer's resignation using a formal letter stating the reasons for the request and presenting it at a meeting of the executive officers.
