It would be better to say "swojej roli".
It would be wrong to say "swojego roli".
Still, it may be someone else's role...
Might be interesting if you put in the Cyrillic versions too: something like "Этот актëр не знает его роли"?
Could you say "ten aktor nie wie jego roli?" I thought znac was used for knowing people personally, not things.
I'd say that "znać" is "to be familiar/acquainted with", and "wiedzieć" is "to have some knowledge". So mostly, "znać" = "to know X", and "wiedzieć" = "to know about X", "to know, that X", etc.
So okay, it might not seem logical for you here, but the actor is 'familiar with his role'. "Nie wie jego roli" makes no grammatical sense, and "Nie wie o jego roli" (Accusative changes into Locative here, but they look the same in this example) would mean "He doesn't know about his role", so that's a different sentence.
Also, remember that "swojej roli" would make a lot more sense here, as it's definitely "his own role" then.
That doesn't seem impossible, but we agreed among moderators that we'd rather phrase the sentence differently then, like "nie zna swojej postaci/swojego bohatera" (more literal choice, exactly to convey the meaning).