the user-interface of Tableau shouldn't depend on whether you have an internal or external connection. The only issue I can imagine right now is caching; the gateway/proxy to the internet might be caching the logos.
Johan de Groot
I don't know the answer offhand, but am adding myself here as a follower.
And how is your TS configured within your network? Is it in the DMZ so it can be accessible by the outside? Tableau doesn't recommend this per this article. Network Security - Tableau