A friend suggested I address this question as a blog post. I told him I would work on it; which means I'll think about it sporadically as it occurs to me over the next day. So, for whatever it's worth, here's my on-the-fly answer: yes, and no. First, let's define "sarcasm". According to Merriam-Webster, sarcasm is a cutting or contemptuous remark or ironic criticism or reproach; and there are examples of Bible saints using this verbal device. One of the most famous is recorded in 1 Kings 18 - the showdown between Elijah and the prophets of Baal. During the competition Elijah says to the prophets about their unresponsive deity, Call out with a loud voice, for he is a god; either he is occupied or gone aside, or is on a journey, or perhaps he is asleep and needs to be awakened (v27). The text calls this Elijah's mocking of the false prophets. His use of sarcasm is neither approved or condemned in the text. It's simply given. Personally, I like it a lot.
The Apostle Paul repeatedly used sarcasm in his letters to the churches, especially the sick churches. For examples see 1 Corinthians 4, 6, & 9.
John the Baptizer used sarcasm. See Matthew 3 & Luke 3.
Jesus used sarcasm. When speaking to the Pharisees, for example, the Lord would often ask them the biting question, Have you not read . . . ? It was His ironic criticism of the "most learned" of Israel not knowing basic Old Testament truth (Mt 12 & 19; Mk 12). He spoke the same way to Nicodemus, a Pharisee himself (Jn 3). I imagine there was sarcasm in Jesus' voice when he asked, Have I been so long with you, and yet you have not come to know Me, Philip? He who has seen Me has seen the Father; how can you say, "Show us the Father"? And the examples could continue. But let me summarize.
Sarcasm, like many devices, is good when used to make much of God and the Word of God. It is not good when used to make much of ourselves and our opinions. By sarcastic comments, people are insulted and shown to be less than they think. When the aim of this is to show someone, with love and sobriety, his ignorance of God, it may be useful. It's a form of correction. But when it is used to exalt self in the presence of others by showing their weaknesses next to our own strengths, it is unloving and selfish and sinful. Blessed is the person with the wisdom to discern the moments when sarcasm is helpful.