These verses contain both a controversy story and a healing, and are unique to Luke. Concerning the passage as a whole, we have the following comments from scholars:
Soards: “This healing story forms the second part of a balanced pair with the foregoing account of Jesus' healing the crippled woman (13:1-16), exhibiting Luke's concern to show the inclusive nature of Jesus' ministry.”
Pohl: “The Gospel of Luke is particularly rich with images of Jesus' experiences of hospitality...He is frequently a guest in the homes of Pharisees (Luke 7:36-50; 14:1-21)...These shared meals provide an important setting for his teachings on divine and human hospitality.”
Luke 14:1-2
Marshall describes the background to this passage: “The situation indicated is doubtless a meal after the service in the synagogue (cf. 7:36; 11:37)...by now the Pharisees were suspicious of Jesus and looking for evidence against him, and it is possible that the man [with dropsy] was planted there as a trap for him.”
Luke 14:2
Regarding dropsy (v. 2), Fitzmyer explains the medical condition as follows: “suffering from edema, an abnormal accumulation of serous fluid in connective tissues or cavities of the body accompanied by swelling, distention, or defective circulation. It is usually symptomatic of more serious problems.” Note that Jesus will alleviate the man's symptoms as an indication that he had also removed the underlying medical cause.
Luke 14:3
Jesus' question showed that He knew that they wanted to trip him up, and in addition the question placed them in a difficult position – for if they answered that it was lawful to heal on the Sabbath day this would give him the right to continue His Sabbath healings. On the other hand, they could not venture to declare that it was unlawful to heal on the Sabbath, especially in view of the distress of the sick man, visible to all.” (Geldenhuys)
I would add that this episode and others imply strongly that Jesus' opponents knew that He had the power to heal all along and didn't seem to reflect on what that meant concerning the authority of His teachings as well.
Another glaring omission committed by the Pharisees was in purposely ignoring the Old Testament teachings on the subject. As Fitzmyer explains, “The attitude presupposed by Jesus in that second question reflects Deut 22:4, the Mosaic injunction to help a brother whose ox or ass falls by the way.”
As Ellis says, “The churchmen display a thoroughly false standard of values. Not only are they hard-hearted toward the sick man (3f.) but both guests (7) and host (12) are status-seekers and social climbers. The principles (5, 10, 13) that Jesus enunciates were known and approved by all. The sting of his words is their utter candor...He undresses their concealed and half-forgotten motives and lays them naked on the dinner table. Most embarrassingly, he applies the principles to the churchmen's relation to God.”
Luke 14:4
Fitzmyer explains that 'they kept silent' indicates ”they would not commit themselves. But to be silent is to agree (especially when legal matters are the issue).”
Luke 14:5
There is a minor textual problem concerning this verse. Comfort notes that if you have a son or ox “is the original wording according to five early MSS [i.e. manuscripts]...Three variants are 'donkey or ox.., 'sheep or ox'...and foal of a donkey or ox'...The variants are scribal attempts to have a pair of animals rather than a human paired with an animal.”
“Jesus highlights their [i.e. oxen's] value when he poignantly equates the Pharisees' sabbath care of their oxen with his own sabbath healing of a crippled woman and a man with dropsy (Lk 13:15; 14:5).” (Dictionary of Biblical Imagery)
Luke 14:6
“Conscious of modernity's anthropocentrism, Christians are noticing Jesus' background assertion that God attends to the death of each sparrow (Matt. 10:29). And they are noticing that Jesus rightly presumes that even his theological opponents will not dare to deny that it is good to break the Sabbath in order to rescue a sheep or an ox from a pit (Matt. 12:11; Luke 14:5).” (Greenway)
Fitzmyer alludes back to v. 4 in which the Pharisees could find no answer to Jesus' question and states, “The reaction [in v. 6] is thus stronger than in v. 4. It is assumed that Jesus' words have reduced his observers to silence. In effect they agree that they would do exactly what he intimated they would.”
Concluding Update
Craddock notes, “It is significant that following the Jerusalem conference, which Luke reports in Acts 15, the Gentile churches are asked to show deference to Judaism only in matters of idolatry, unchastity and foods (Acts 15:20). No mention is made of the Sabbath. Either the matter had already been settled, or it was not a critical issue, or it was left to the convictions of different Christian communities to observe or not observe the Sabbath...One suspects, however, in Luke's time and place, that...Sabbath observance still generated a great deal of tension.”
As a result of such actions and teachings of Jesus, “the extreme (and potentially legalistic) forms of refraining from work that prevailed under the law are softened and set within the context of God's new redemptive work...Given this NT modification we are left with a picture of the sabbath as a day of doing good as well as a day of worship and cessation from ordinary work.” (DBI)