If one of our sources of authority is the examples set for us in the New Testament, and since Paul talks about the first century church fasting, why don’t Christians fast today?

Are you sure that Christians DON’T fast today, as in at all, ever?  After all, consider Christ’s directives about fasting in the Sermon on the Mount:

Matthew 6:16-18 (ESV)
¹⁶ “And when you fast, do not look gloomy like the hypocrites, for they disfigure their faces that their fasting may be seen by others. Truly, I say to you, they have received their reward.
¹⁷ But when you fast, anoint your head and wash your face,
¹⁸ that your fasting may not be seen by others but by your Father who is in secret. And your Father who sees in secret will reward you.

In the context of this passage, Jesus was rebuking the hypocrisy of the Jewish leaders of his day (Matt. 6:1ff).  They let everyone know they were fasting because they wanted the praise of men.  Jesus wanted his followers to be the opposite of that.  When they fasted, Jesus wanted them to keep it between themselves and God.  He wanted it to be the case that anyone who concluded that they were fasting would have had to have done some detective work to figure it out.  This way, only God will see you fasting and he will reward you openly (perhaps by answering those prayers you were offering while you were fasting.)

Therefore, if it looks to you like no Christians anywhere are fasting, that may just be because more people in the church are taking Jesus’ directives here to heart than you might think.

Jesus DID command fasting to be done.  That’s clear by noting that he said in the above passage, “WHEN you fast…”, not “IF you fast…”  Additionally, Jesus himself fasted (Lk. 4:1-2), as did Paul (2 Cor. 11:27).  We are to imitate him, just as he imitated Christ (1 Cor. 11:1).

It’s worth noting the various reasons why fasting took place in the Bible.  The early church fasted and prayed when they sent out missionaries (Acts 13:1-3) and when they appointed elders (Acts 14:21-23).  In the Old Testament, Israel fasted during unusual times of stress like impending or occurring war and they wanted the protection of God (Judg. 20:26; Ezra 8:23).  David fasted when his child was sick and he prayed for God to heal him (2 Sam. 12:16).  Daniel fasted when he wanted God to forgive him (Dan. 9:3).  Interestingly, so did wicked king Ahab (1 Kings 21:27).  Jesus fasted while dealing with the temptations of Satan (Lk. 4:1-2).  The Psalmist fasted while praying in order to become more humble (Ps. 35:13; 69:10).

Along these lines, it’s also noteworthy that there is, as far as I can tell, no formal regular ritualism directed to us concerning fasting.  In other words, the New Testament (the covenant which we are under today) does not tell us to fast for so long on such-and-such days for such-and-such specific reason on a weekly, monthly, or annual basis.  Rather, it seems that God considers fasting something good for Christians and churches to do when faced with serious, unusual situations that require his help, like difficult temptations, the serious illness of loved ones, appointing elders, starting new works, etc.

From what I can gather, fasting is a good way to humble oneself in the sight of God while seeking a favorable answer to one’s pleas to his throne which are lifted up to him in prayer.  Exactly how, when, and how long one does it, as well as the fact that it is happening at all, should be between him and God.