Tartar builds up where the tooth meets the gum. It is hard, mineralised plaque that contains bacteria. If it is not removed, it can cause inflammation and, over time, lead to gum recession.
Moving teeth while there is inflammation is not recommended, so tartar should be removed whenever necessary. Dentists have cleaning methods that do not damage braces or cause them to come off. How often cleaning is needed depends on the individual patient.
With aligner treatment, once the aligners have been taken out, the dentist has full access to remove the tartar. It is recommended that the aligners are cleaned very thoroughly with disinfecting tablets before they go back in.
This answer is for general information. Your own case needs an examination by an orthodontist.