No, the IP Address won't be updated, generally speaking. If you are using a different membership module besides the one built in to ExpressionEngine, then maybe. But generally speaking, ExpressionEngine (EE) logs the IP Address the user was at when they created their account or when they are logged in, but it doesn't require the same IP Address.
I can or you can log-in from any ip_address, unless you've put blockers in place to restrict to specific IPs. The control panel option for tracking ip_addresses reads:
Require IP Address and User Agent for Login? Prevents users from
logging in unless their browser generates IP Address and User Agent
data. This keeps hackers from logging in using direct socket
As it states, it prevents
socket use, and while it does log current IPs, it doesn't do member matching. At least not to my knowledge. Checking my Control Panel logs, I can see that I've logged in from at least 7 different ip_address, though the IP Address on my account profile, remains my home ip_address.
Now, if you are using Forums, this maybe different. We've never used EE Forums for clients, so I can't comment. But generally speaking, the
ip_address is used for basic security, like you could ban a country or a block of IPs, that have been problematic.
So you should be fine, unless you have other restrictions by ip_address in place.