You usually need a good reason to remove a trustee, such as if they have done something that damages your charity’s reputation.
If your charity is a company, you have the right to remove a director, providing you follow the correct procedures. You have this right under the Companies Act 2006, regardless of what else is written in your articles of association.
You can hold a vote of no confidence to encourage someone to resign as a trustee. This could be part of your charity’s rules for removing a trustee, or written into its governing document. If it isn’t part of your charity’s rules, the vote has no legal power and the trustee won’t have to resign.