Need steps to reboot Big IP LTM in HA pair without impacting production?

It's also worth noting above - To give your business full confidence of the procedure. You may wish to conduct a "Failover" without the reboots prior to the main event. I.e.

Failover from Active to Standby Test applications (Buisness Critical) for a good period. Failover from Active to Standby again.

I've known scenarios in the past where the standby box self IP's for health monitoring haven't been added to internal firewalls etc, which may only manifest when in a failover scenario.

It's also an opportunity to test how the applications behave, as a fallback is a lot faster to resolve than waiting for a box to fully come back up.