I agree that the feeling of immersion is at stake here (I feel this is why it was so off-putting to see Ed Sheeran in GoT scene), but I think this actually depends on the actor. There's so many that lack the talent or the will to mix it up, and consequently just playing themselves in roles. Chris Evans, Megan Fox, Jennifer Aniston, Jason Statham are some examples that come to mind.

Then there's others, like Daniel Day Lewis, Di Caprio, Ralph Fiennes, etc. who's roles comes across as being fiercely distinct and yet still utterly convincing.

This is a total paraphrase and possibly a mis-creditation, but I'm pretty sure Ian McKellen said something along the lines of "When you watch good actors, you think, they're a good actor. When you watch great actors, you actually believe they're a real person."