Likely reasons Dan omitted
One standard speculation for why Dan is omitted comes from the fact that they had an especially great problem with idolatry. The tribe was early devoted to such practices (in fact, the first tribe to do so) and continued them up to the time of their captivity in 723 B.C. Its worship of idols began right after they fought for and secured their inheritance in the promised land:
"In those days there was no king in Israel. And in those days the tribe of the Danites was seeking an inheritance for itself to dwell in; for until that day their inheritance among the tribes of Israel had not fallen to them. So the children of Dan sent five men of their family from their territory, men of valor from Zorah and Eshtaol, to spy out the land and search it . . .
"And they called the name of the city Dan, after the name of Dan their father, who was born to Israel . . . Then the children of Dan set up for themselves the carved image; and Jonathan the son of Gershom, the son of Manasseh, and his sons were priests to the tribe of Dan until the day of the captivity of the land. So they set up for themselves Micah’s carved image which he made, all the time that the house of God was in Shiloh." (Judges 18:1-2, 29-31)
Later in its history their idolatry was further encouraged when they, along with nine other tribes, split from a united Israel after the death of Solomon. The newly formed Northern Kingdom was ruled by King Jeroboam. Right after he took power (in 930 B.C.) he became afraid his new subjects would travel to Jerusalem to worship God, repent of splitting from their brethren, put themselves back under the rule of Solomon' son and kill him. His solution was to set up false gods within the boundaries of his kingdom to keep the people separate from their brethren in the Kingdom of Judah. The first idol was placed in the city of Dan:
One theory sometimes discussed is that Dan is not listed because the Antichrist will be a Danite. There is no known credible Biblical proof, however, to back up this claim.
"And Jeroboam said in his heart, 'Now the kingdom may return to the house of David: If these people go up to offer sacrifices in the house of the Lord at Jerusalem, then the heart of this people will turn back to their lord, Rehoboam king of Judah, and they will kill me and go back to Rehoboam king of Judah.' Therefore the king asked advice, made two calves of gold, and said to the people, 'It is too much for you to go up to Jerusalem. Here are your gods, O Israel, which brought you up from the land of Egypt!' AND HE SET UP ONE IN BETHEL, AND THE OTHER HE PUT IN DAN." (1Kings 12:26-29)
A likely contributing factor to their omission is that they, situated on the northern border of the area originally given by God to Israel, were very slack about driving out the pagan Canaanites in their midst.
The tribe of Dan also didn't always participate in the collective defense or offense of Israel as they should have. When Deborah sang about her victory over the Canaanite king Jabin, she asked, "And why did Dan remain on ships?" (Judges 5:17). But then, admittedly, Asher was also slack in the same battle according to this same verse, yet gets mentioned in the book of Revelation. As a sidelight, the most well-known Biblical person from the tribe of Dan was Samson. Although he was a judge of Israel for twenty years and known for his miraculous strength, he was not a very spiritual man for most of his life.
Ultimately, the real issue is that those who believe in Scripture, when confronted with mysteries like this, still need to live their lives rightly and to have the right faith to ensure they are considered worthy to enter God's kingdom. May we strive to be there!