In Star Wars - Episode IV: A New Hope, R2-D2 is winning a game against Chewie, and Chewie got angry. C-3PO said R2-D2's move was fair, and Chewie shouldn't have complained. Han told him to let Chewbacca have it, because
It isn't wise to upset a Wookiee.
C-3PO points out that no one worries about upsetting droids, to which Han replies:
That's cause droids don't pull people’s arms out of their socket when they lose.
But Chewbacca is playing R2-D2, and R2-D2 doesn't have arms.
According to the original novelization of Star Wars: A New Hope, credited to George Lucas: Both R2-D2 and C-3PO.
Attracted by the commotion, Solo looked back over his shoulder, frowning slightly. “Let him have it. Your friend’s way ahead anyhow. It’s not wise to upset a Wookiee.”
“I can sympathize with that opinion, sir,” Threepio countered, “but there is principle at stake here. There are certain standards any sentient creature must hold to. If one compromises them for any reason, including intimidation, then one is abrogating his right to be called intelligent.”
“I hope you’ll both remember that,” Solo advised him, “when Chewbacca is pulling the arms off you and your little friend.”
“Besides that, however,” Threepio continued without missing a beat, “being greedy or taking advantage of someone in a weakened position is a clear sign of poor sportsmanship.”
That elicited a beep of outrage from Artoo, and the two robots were soon engaged in violent electronic argument while Chewbacca continued jabbering at each in turn, occasionally waving at them through the translucent pieces waiting patiently on the board.
- Star Wars - Episode IV: A New Hope, George Lucas (ghost written by Alan Dean Foster)
This confirmed in Legends, specifically in the formerly canonical novel Shadows of the Empire. Chewbacca and C-3PO are playing the holo game when this passage takes place:
Chewie said something that sounded angry and impolite.
Threepio said, “Well, I’m sorry, but it was a fair move. It’s not my fault you didn’t see it.”
Chewie said something else.
“No. I’m not going to take it back. And don’t threaten me. If you pull my arm off, I won’t play with you anymore.”
Chewie muttered something, then leaned back on his seat and looked at the game board.
Note: I don't like answering my own questions, much less accepting my own answers, but I just stumbled upon this and had to post it as an answer, since it is effectively Word of God. In order to prevent myself from gaining rep from this answer, I've made it Community Wiki.
Han Solo never indicated that Chewbacca would rip anyone's arms out.
If you look at the way in which the conversation is setup, Han is never specifically referring to Chewbacca. Instead, Han gives a general comparison between wookiees and droids, remarking that wookiees are known for pulling people's arms out of their sockets while droids, obviously, are not.
Han is using this example of "wookiee behavior" to demonstrate that Chewy could be violent should he loose.
Still, Han's argument is directed towards C-3PO based on C-3PO's reaction to Solo's threat.
Han: Let him have it. It’s not wise to upset a Wookiee.
C-3PO: But sir. Nobody worries about upsetting a droid.
Han: That’s cause a droid don’t pull people’s arms out of their sockets when they lose. Wookiees are known to do that.
C-3PO: I see your point, sir. I suggest a new strategy, R2. Let the Wookiee win.
It has been established in this answer that droids can feel pain. C-3PO was in a position to direct R2-D2's gameplay, given by the line regarding the "new strategy" and C-3PO was the only droid with "arms." Thus, Han Solo must have been directing his comment to C-3PO.
Luke was practicing with his lightsaber and Ben was not involved in the game. By method of deduction, it must have been C-3PO.
Han Solo wasn't saying that if R2 should win, Chewy would rip his arms off (but even if he was, that sounds much better than any droid part you can name). Rather, he was probably saying that Chewbacca can get angry if he loses, and that when Chewbacca is angry, he has been known to rip people's arms off. If Chewbacca had lost, he would not have ripped off any arms, but he might well have ripped off something or thrown R2 around the room. (This wouldn't work because R2 would electrocute him or something, but Han did not know of R2's awesomeness at the time).
In the script for The Force Awakens, it is revealed that Chewie would have ripped Unkar Plutt's arm of.
