Why Tails or Whonix? Running Bitcoin Core (which is needed for Armory) in either Tails or Whonix is a pain in the ass - the blockchain is simply too big.
I'd say the best option is to run a standalone Bitcoin Core (not managed by Armory) over Tor - you just need the Tor Browser Bundle to do that.
In Bitcoin Core: go to Preferences/Network and check "connect through SOCKS5 Proxy". Proxy IP: 127.0.0.1, Port: 9150 - and then have Tor Browser Bundle open together with Bitcoin Core.
After the above, in Armory just check check "enable settings for proxies/Tor" in Privacy Settings.
The issue with connecting to a anon/private network through a proxy is that any traffic that doesn't go through the proxy can reveal you. As long you use something like the Tor bundle on top of a regular system, that remains a possibility. The main offenders in these setups are DNS lookups: regular traffic will go through the proxy, but your machine will be sending domains in plain text to your ISP for resolution.
It is fairly hard to sanitize an application/system to avoid this kind of (otherwise) routine and harmless side channel traffic. With an OS like Tails or Whonix, that problem goes out the way.
The other point is your OS spying on you. That's a given for Windows so let's not touch on this topic here. For Unix flavors, while the purpose isn't actually spying, there is a wealth of phone home code that will ignore the Tor proxy.
From my perspective it's easier to keep Bitcoin and its blockchain data in an encrypted removable disk for using on Tails than to try and sanitize my regular Debian install. Also, AFAIK Tails is the live Tor OS whereas Whonix is the perma install one. It would be an even easier setup with Whonix.
