At the lowest level, they are completely unrelated. In a practical sense, methods exist for assigning IP addresses based on MAC addresses, but your ISP almost certainly has assigned your IP address to your physical line, and changing your MAC address is unlikely to have any effect on your IP address.
Changing your MAC address is as easy as buying another ethernet card and substituting it. For a PC, that makes it pretty trivial. For a router, it's as simple as buying another router. But neither of these will have any detectable effect outside your local net.
-- George Mitchell (obfuscated email address)