I have the paper backup for Armory Wallet but do not have access to the computer which the Wallet was created. How to I export the BTC?
You can just import the backup on another computer with armory installed. It's not dependent on the computer it was created on.
Remember: The coins don't actually reside on your computer - The public ledger called "blockchain" holds the amount you own. You rather hold the private key in your wallet app, that allows you to sign transactions and thus send money.