Sites do not tend to block using the public IP address for a number of reasons:
1) Most home users have dynamic public addresses and tomorrow, you might have a totally different address. Somebody else might get your previous address then they would be blocked instead of you.
2) There are many ways of actually getting around an IP address block. The main one is probably through the use of a proxy server.
3) If there are several people in a building using the same ISP connection, then they will all be sharing the public IP address. If they block that address, then everybody in the building will be blocked.
Generally, you cannot simply start changing addresses and expect the network to continue to work properly. If you have to ask a question like this, then I am sorry to say that I doubt if you understand enough about how networking works to make any changes safely.
Most site blocks are performed by one or more of the following:
1) If you have an account at the site, then the account is locked.
2) The site can place a cookie on your computer that says you cannot access the site.
3) If there is a specific application that you run to access the site (in this case, perhaps you are running a Minecraft application), then they can set a marker in the application that blocks you.
4) They can actually recognise the specific computer and block any access from that computer. Microsoft use this approach to prevent installing Windows and applications on more systems than the licence allows.
5) If you register a credit card on the site, then the card number can be blocked from using the site.
A site might use several of these methods to block you. If you change you IP address, create a new account and register with false details and a different credit card, if the site recognises the computer, then it can block the new account and the new card. The new IP address will probably be ignored as explained above.
You have to change everything to be sure of removing the block.
I hope this helps.