You need to edit the MX records in the DNS Server which is hosting the DNS for your company. This could be your own Public DNS server, or that of your web host.
If the public IP Address of your DNS server did not change then you do not need to make any changes to the MX record.
Depending on who is hosting your DNS server, and what kind of DNS Server it is (Microsoft, BIND, etc) the process to edit the files will be different.
MX records point to an A record. The A record will need to be modified with the new IP address or create a new A record and point the MX record to the new A record.