Pip does not want Drummle to see Joe because he is ashamed of Joe’s simple manners.
Drummle, described by Jaggers as the “blotchy, sprawly, sulky fellow” (ch 26), is Pip’s nemesis. Drummle offends him and irritates him. He is a perfect stereotype of the spoiled gentleman: brutish, boorish, and mean.
I had little objection to his being seen by Herbert or his father, for both of whom I had a respect; but I had the sharpest sensitiveness as to his being seen by Drummle, whom I held in contempt. (ch 27)
Pip is aware that he is ashamed to show his uncle to the man he despises, and ashamed of that as well. Joe is a simple blacksmith, not a gentleman. He has crude manners and clothes, and is not comfortable in the city’s upper circles.
His shame in Joe reveals something deeper about Pip. Joe is a symbol of Pip’s humble upbringing. He is a reminder that just as everything was given to him quickly and unexpectedly, it can be taken away. Pip is tormented by the idea that he might one day turn into Joe.
The fact that Pip would be willing to let Herbert and his father see Joe is an interesting one. It shows that he respects them as his equals, not his betters.