Any user account you have on your system... the ones that have PSN accounts attached to them are most important. All you do after you install the new board is make new user accounts for each PSN account. Then, go into each user account and just sign into all of your PSN accounts. Anything that you do online is always connected to your PSN account, so yes. MW3, along with any online game requires your PSN account to retrieve your progress. If you have DLC or any other games/content downloaded from the PlayStation Store, your PSN accounts will be essential to retrieve those as well.
This is why you must know the credentials for all of your accounts if you have more than one. Or, you could just remember the credentials of the accounts you want to keep. If you have bought items from the PlayStation Store, they will be attached to whatever account you bought those specific items from. If you have bought things from several accounts, then you will need all of your accounts if you wish to re-download your purchased content.
Also, if you have substantial online progress on a certain account, just make sure to get the credentials for that account. After you sign into the account, your games will fetch your online stats and progress. IF you have other accounts with good stats, just do the same for those as well.