If we examine the instances when Robert performs this curious behaviour, we can see that this gesture is something that indicates Robert's need to reassure himself that he is OK. The beard is something that the narrator focuses on from the first time he sees Robert, saying: "The blind man, feature this, he was wearing a full beard! A Beard on a blind man! Too much, I say." However, this beard seems to be something that Robert touches to reassure himself. Note the way that we are told how he touches his beard before addressing the narrator:
From time to time, he'd turn his blind face toward me, put his hand under his beard, ask me something.
Given the less than hospitable welcome the narrator has given Robert, we can therefore read this gesture as one that indicates Robert's nervousness and shows his attempt to reassure himself.
At the same time, later on in the story, it seems as if this gesture is indicative of deep thought and reflection. Consider what the narrator tells us about how Robert touches his beard when they are watching TV together:
Now and then he put his fingers into his beard and tugged, like he was thinking about something he was hearing on the television.
The use of Robert's beard seems to be another way of expressing the emotions and feelings of Robert as a character. Because the story is told exclusively from the point of view of the narrator, Robert's beard and what he does with it therefore gives us an idea of what Robert is feeling and experiencing, as the narrator shows himself to be too self-absorbed on the whole to realise this.