Do I need to install Armory on a computer to move my BTC out of an armory wallet?
Yes. To move BTC OUT of the wallet you need to access the wallet and issue the send command (type in password if you set one) to send coins to another address.
You can send coins to your paper backed up address without having that wallet online, even if you created the wallet offline and it never touched the internet once.
This is a security design to keep your coins as safe as possible. Otherwise anybody else could just move your coins.
If you lost your keys or passwords then the coins are lost, even if you know the address.
which keys are you talking about. There are no keys private or public in bitArmory. It is impossible to get private keys ,unless someone else knows how