What is the significance of Becket's silence after the fourth temptation?
I am assuming you are referring to the play Murder in the Cathedral, not the Jean Anouilh’s version Becket. In this play, Thomas Becket does reply to the fourth tempter - after a short hesitation. This tempter is tempting him to commit what he believes is a grievous sin. To seek to become a martyr in order to gain personal glory is the worst example of pride a man can have. Becket believes that if he is to be a martyr, it is by God's will. He tells the tempter:
The last temptation is the greatest treason: to do the right deed for the wrong reason.
The fourth temptation, therefore, would be the worst sin in Becket's mind. He does not care for his physical safety (first temptation), he has already served the king and has power (second temptation), and a coalition against the king would be political treason which he does not want to commit. But treason against God would be the worst.
If this is not what you are asking, you can read the analysis for both Becket and Murder in the Cathedral here on eNotes.
The archbishop is silent after the fourth temptation because it really does tempt him. The fourth temptation is to do the right deed for the wrong reason. Becket knows he should stand firm against the king solely because it is the right thing to do, morally and ethically, to resist a tyrant, but he is tempted to stand firm out of a desire for martyrdom. He wants to go down in history, remembered as dying for his glorious deed. Becoming a martyr however, merely for the sake of glory is a form of pride that can lead one to throw one's life away at the wrong time. The true martyr has given up his own will to become entirely an "instrument" of God. Beckett has to get his head together and sort out the bigger picture of exactly why it is he is taking on the momentous task of opposing his king. He has to be clear he is obeying God and not his own desires. It can help to know that the Bishop of Chichester commissioned this play in the 1930s to urge people to stand firm against another tyrant, Adolph Hitler, but to do so for the right reasons.