this is one of those times where reading the EULA is handy... You know.. that big text document it shows when you goto play online...

If you are talking about a LEGITAMATE backup, I think it will work. However if your talking about a copy of a friends game or something you got off the internet I don't think it will. Each copy of the game has a unique serial number that is checked when you go online and matched up with another unique number for your PS2. They use this for tracking win/loss records and such. So if it really is just a backup you should be fine as nobody else will have the same number. If its something you DLed and 1000 people are using the same "copy" it won't work. I think.

DONT DO IT MAN THEY WILL BAN YOU!

Do it. Microsoft doesn't mind warez on Xbox Live. You will have no problems at all, and shouldn't get banned.

I don't know how PS2 system works for online, but i know for a fact that you cannot use backup games on Xbox Live!
You cannot even use legitimate imported games on Xbox Live. To use both of these requires a modchip. If your modchip is switched on when you log onto Xbox Live you will automatically be banned from the servers. Microsoft detects a modified BIOS on your Xbox.