Of course he could have a major impact. As curly mentioned, he could fill the Dave Roberts role and as we all know, that role can have an enormous impact on a game and a game can be the difference in a postseason series.
I see two scenarios where he has a significant impact.
1) He's on the post season roster as a supersub and either steals an important base or manages to come up in an important spot because the NL sucks.
2) He continues to destroy AA through June, comes up to AAA and destroys it in July and August, comes up in September and excels in limited time before someone gets hurt, forcing him into a more prominent role than anyone really anticipated.
Regardless, I want him up in September and actually, now that I think of it, he could certainly have an impact in a critical game if the Sox are in a dogfight. Just to imagine one scenario, the Sox are playing at the Yankees, tied in the standings, Lester against Tanaka, Betts plays left because he's got better range there than anyone else we could put there. He makes a catch or two that Gomes wouldn't get and if it's in the right situation, it could be huge.