MS SQL Server IP address change considerations
If you are just changing IP address ONLY, then there are less things to worry.

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Change the IP address and restart SQL Server.
If your applications are referencing IP address instead of HOST, then they need to change their web.config files to reflect the new IP address.
Linked servers referencing IP address needs to be changed as well.
If any user workstations are connecting using IP, then they need to flush DNS using ipconfig /flushdns or better just restart their workstations.
If the server is part of Failover cluster then follow this steps as well.


If you want to be proactive and find out the applications connecting to your sql server instance, then use this server side trace and then plan accordingly for downtime.

Note: make sure you assign a static IP address and port to sql server.