Asia’s run out of IPv4 Addresses

tl;dr

Simply stated, how will this affect the regular world if at all?

How would running out of telephone numbers effect the real world? Same issue, impossible to give the exact ramifications because every household & company uses a lot more IP addresses than phone numbers for a much wider array of communication needs. You can't just tack on an extra 4 digits to every phone number and call it solved because every company, every device, every person has to know the new numbering system and where they fall in it and with IP addresses that's 30 years worth of software written to do any number of things.