I did some more searching and I have found a number of posts with a similar problem where the Mac will shutdown after running for 4 to 10 seconds unless you press and hold some key (or as you have stated "any" key). I am not seeing many desirable solutions and the posts are indicating that something is wrong with the keyboard cable and/or the power button. The problem is supposedly solved by replacing the keyboard and its related components. Others have attempted taping down the eject key or the fn key to keep the shutdown from occurring as there appears to be no negative side effects of having those keys pressed. Still - this is not a very good solution.
Others have suggested trying to use "sudo pmset powerbutton 0" or "sudo -c pmset power button 0" (The -c only sets this option for when the Mac is running on the AC Charger). Important note" the "powerbutton" parameter is only recognized by pmset in 10.6.x - and has been removed as of Mavericks (or earlier). If this provides an improvement - it will not in later versions of OS X.
Some have suggested setting the keyboard repeat rate to "None".
To find these posts I did a search in Safari on "2010 White MacBook Unibody Shuts Down unless a key is pressed".
You could attempt to disassemble the MacBook and maybe reseat any cable attached to it. I have never taken a MacBook (or Mac portable) apart other than to upgrade RAM. It is my understanding that the internal components have a very tight fit and must be removed in a specific sequence to avoid damaging some fragile cabling.
You did not indicate whether the MacBook also shuts down when booted up from OS X Installer DVD - or when running the extended (long running) diagnostics via the D key at boot up?
Have you checked to see if there are any firmware updates available - either via Software Update or via direct download from Apple?
Another you could try as a troubleshooting measure (which I doubt will produce any definitive results) would be to clone you boot volume onto an external drive and attempt to boot up from the external drive. To clone your boot volume you would boot up from the OS X Install DVD - and use disk utility to "restore" your boot volume to a partition on an external drive. You would then be able to boot from that.
~Scott