There is an argument for Jesus not being born on December 25th. One of the arguments runs like this:
During the time of King David, there were so many priests, that they had to be divided into 24 'courses' or groups. Each course would serve in the Temple for a period of 1 week, starting with the first Sabbath of the year (Month of Nisan). During the 3 annual Holy Days, when all of Israel had to gather in Jerusalem, all the priests would serve together, then resume the group rotation the following week. Since Zechariah belonged to the 8th group, he would be in the Temple during the 9th week of the year. ..."
This would put Zechariah's turn to serve in the Temple during the month of Sivan. When he completed his week of service, he returned home. It is reasonable to assume that Elizabeth became pregnant later that month, (Sivan).
John the Baptist was then born 40 weeks (9 months) later, during the month of Nisan. Since John was about 6 months older than Yeshua (Luke 1: 36), The birth date of Yeshua must be during the month of Tishri.
The lunar month of Tishri occurs during our September or October. (See this site)
Another argument by Ray Butterworth runs like this:
The Fall makes far more sense than late December...
Elisabeth conceived (John the Baptist) immediately after Zacharias finished his Abia priestly duties (Luke 1):
 ... priest named Zacharias, of the course of Abia ...  ... he executed the priest's office before God in the order of his course  ... as soon as the days of his ministration were accomplished, he departed to his own house.  And after those days his wife Elisabeth conceived ... Her cousin Mary conceived (Jesus) six months later:
 And in the sixth month the angel Gabriel was sent from God unto a city of Galilee, named Nazareth  To a virgin espoused to a man whose name was Joseph, of the house of David; and the virgin's name was Mary.  And, behold, thy cousin Elisabeth, she hath also conceived a son in her old age: and this is the sixth month with her, who was called barren.  And [Mary] entered into the house of Zacharias, and saluted Elisabeth.  And Mary abode with her about three months ... The course of Abia is defined in the First book of Chronicles:
[24:10] The seventh to Hakkoz, the eighth to Abijah [24:19] These were the orderings of them in their service ... Each priest served for a week, with the cycle beginning on the 1st day of Nisan. But during the week of Passover (Nisan 14) and Pentecost (7 weeks later) all priests served together, so Zacharias, being of the 8th course, would have finished his first course after the 10th week of the year. Six months after that (Jesus's conception) would have been around what the Romans called late December, and nine months after that (Jesus's birth) would have been late September or early October.
That would be around the time of Sukkot (the Feast of Tabernacles). Sukkot is one of three times each year when people made a pilgrimage to Jerusalem. In addition to the normal tithe given to support the priesthood, people were encouraged to save a second tithe to use for their own expenses and celebrations during these times (Deuteronomy 14:26):
And thou shalt bestow that money for whatsoever thy soul lusteth after, for oxen, or for sheep, or for wine, or for strong drink, or for whatsoever thy soul desireth: and thou shalt eat there before the LORD thy God, and thou shalt rejoice, thou, and thine household. Even today, most people that observe this fall festival travel away from home, or if they can't, they arrange to sleep in a tent or some other location different from their normal home.
It's by no means a certainty that this is when Jesus was born, but if Jesus had been born during the Feast of Tabernacles, it then confirms why his parents stayed in their family's home town of Bethlehem, just south of Jerusalem.
How would those who believe in the traditional December 25th date of Jesus birth answer these arguments?
What are the earliest references to Jesus' birthday being December 25?
Why would the Shepherds be out at night in the winter during Jesus' birth?
According to the Jehovah's Witnesses, why was Jesus likely not born in December because of Levite temple service schedule?