You need to have dedicated Postmaster and Abuse email addresses, or role accounts, for each sending domain. RFC 5321 states that errors during the processing of email should be directed to the Postmaster email address. Likewise, RFC 2142 indicates that any abusive behaviors such as spamming should be reported to the Abuse email address.
Mailbox providers may also try to communicate to you directly by sending an email to either the Postmaster or Abuse role account. Without these role accounts, important communications may be missed.