If you are referring to the SMTP client sending limits ( does not apply to Outlook Web Access or Outlook in "Exchange" mode), then if that limit is exceeded, the messages over the limit are rejected:
Message rate limit: Message rate limits determine how many messages a user can send from their Exchange Online account within a specified period of time. This limit helps prevent over consumption of system resources by a single sender. If a user submits messages at a rate that exceeds the limit via SMTP client submission, the messages will be rejected and the client will need to retry.