Diplomatic Support and Security Act of 2021 This bill modifies procedures for investigating serious security incidents related to a U.S. government mission abroad and addresses related issues. Under this bill, a serious security incident involves (1) the loss of life, serious injury, or significant destruction of U.S. government property abroad; or (2) a serious breach of security involving intelligence activities of a foreign government directed at a U.S. government mission abroad. Currently, accountability review boards convened by the Department of State investigate and report on these incidents. The bill establishes the Serious Security Incident Investigation Permanent Coordinating Committee to determine whether an event qualifies as a serious security incident. The State Department Diplomatic Security Service must assemble a team to investigate certain events relating to a U.S. government mission abroad that may be a serious security incident. The assembled team must prepare a report for the coordinating committee about the incident that includes an accounting of the relevant facts and an assessment of applicable security procedures and countermeasures. The coordinating committee must review the investigation report and make findings about, for example, the adequacy of security measures. The committee must provide, within 60 days, its findings and related recommendations to the Secretary of State. The findings and related recommendations must also be shared with Congress. The bill also requires the State Department's monthly briefings to Congress on embassy security to include an assessment of the national security risks stemming from the continued closure of high-risk, high-threat posts.