Why does the holographic droid act so weary and 'faint' whenever he finishes delivering a message from Vader to Starkiller? He never acts like that when projecting other things.
PROXY "hates" impersonating Vader:
The disgust PROXY felt at being forced to impersonate such a hated individual is probably strenuous and makes him weary.
PROXY already had issues with the Sith Lord, as he found impersonating him during his frequent communications with Marek highly strenuous, and his realization about the brutality of Vader's methods left PROXY disgusted by him.
It seems that by projecting Vader, Proxy absorbed some of Vader's extreme negative emotions.
Proxy (on Vader): I hate being him.
Now, it doesn't really make sense why projecting Vader is emotionally draining, while projecting similarly angry beings such as Maul isn't, but it's the best info we've got.