"I'm going to swap a dying drive with a new one, will the out-of-date updates have a negative impact when I swap the drives - hot swap?" Unlikely.  Hot-swap is a basic function of the hardware, so it will be supported regardless of the firmware versions, however, there are rare instances where firmware bugs can lead to unexpected behavior, which firmware updates are often issued to fix (which is why firmware updates on servers are important).  It is extremely unlikely that the firmware version will affect swapping the bad drive, and you don't want to update controller drivers/firmware when it's "degraded", but you should plan to update the system firmware/drivers once everything is back up and healthy again.
I have OMSA installed on the server and stated firmware and drive update is out of date. How do I go about updating this? Will this have any affect if I do a hot-swap on a drive? Thanks!
PowerEdge R710 Firmware/Driver Update