Do Christians need to keep the Passover on the 14th day of the first month? If not, why?
No, paticularly if you are not a Jewish (Messianic) Christian. The outcome of the Council of Jerusalem is quite clear - Gentile (non-jewish) believers do not need to follow Jewish customs or ceremonial law with the following caveat:
...It seemed good to the Holy Spirit and to us not to burden you with anything beyond the following requirements: 29 You are to abstain from food sacrificed to idols, from blood, from the meat of strangled animals and from sexual immorality. You will do well to avoid these things... (Acts 15:28-29 NIV)
In the case of Messianic Jews, there is no necessity to keep the Passover to satisfy a legal requirement (Christ has completely fulfilled the ceremonial law), but it is quite legitimate for them to observe the Passover out of a desire to preserve their Jewish culture and to identify with their fellow Jews - to which the Apostle Paul gives them a good example and relevant instruction:
To the Jews I became like a Jew, to win the Jews. To those under the law I became like one under the law (though I myself am not under the law), so as to win those under the law. (1 Corinthians 9:20 NIV)
The short answer is no, because Christ fulfilled the Passover. He is the true Passover Lamb, the "Lamb of God, who takes away the sin of the world" (John 1:29, cf. Revelation 5:6).
In the Old Testament, the Passover was instituted the night the Lord led His people Israel up out of captivity in Egypt. The blood of the Passover lamb was placed above the door frames and post of the homes of the Israelites to mark them as the Lord's people. This mark of the blood of the lamb caused the angel of death to "pass over" the Israelites, saving their first born sons from death (Exodus 12:21ff).
This was pointing forward to what Christ would do on the cross. He came as the true Passover Lamb, marking us with his blood, and causing eternal death to "pass over" us, thereby freeing us from captivity to sin and death. In fact, just as the Israelites ate the Passover Lamb, so too does Christ institute a Supper for himself; in the Lord's Supper we eat the body and blood of Christ which was "given up" and "poured out" for us (cf. Matthew 26:26; Mark 14:22; Luke 22:19; 1 Corinthians 11:24).
So, just as Christ fulfilled the Sabbath for us (by resting in the tomb) and all the commandments by being perfectly obedient, even unto death (Philippians 2:8), so too does he fulfill the Passover, completing what it pointed to.