Good question!
Shi'i texts give the following dates on the Islamic lunar calendar for the birth of Jesus:
(a) 10 Muharram. From the chain of narration from 'Ali ibn al-Hasan, from Muhammad ibn 'Abd Allah ibn Zurarah, from al-Bizanti, from Aban ibn 'Uthman, from Kathir al-Nawa', from Abu Ja'far [al-Baqir] (A). He said: "The Day of Ashura is the day that Jesus the son of Mary (A) was born on." [Bihar al-Anwar, vol. 14, p. 206] <- Personally I think this is less likely, but it's there.
(b) 25 Dhu al-Qa`dah. From Ibn al-Walid, from al-Saffar, from Ibn 'Isa ibn Hashim, from al-Washa', from [Imam] al-Rida (A). He said: "The night of the 25th of Dhu al-Qa`dah is when Ibrahim (A) and Jesus (A) were born." [Also Bihar al-Anwar, vol. 14, p. 206]
Historians say that Jesus was born around 5 BC, which would correspond to -645 / -646 AH. So, this would correspond to either 22 October (10 Muharram -645 AH) or 8 September (25 Dhu al-Qa`dah -646 AH).
If one takes 1 AD as the date of birth of Jesus (since the year 0 AD is not used in the calendar), it would be August 20th (10 Muharram -639 AH) or July 7th (25 Dhu al-Qa`dah -640 AH).
In either case, the resulting date is not 25 December.
That being said, I don't think there are any historians who argue that Jesus was actually born on 25 December; it is just the time of the year that the birth was eventually commemorated on. Also, the specific date on which Christmas has been celebrated has historically varied between countries.
(Disclaimer: It is possible that these calendarical conversions are somewhat off since there have been some adjustments to the Gregorian calendar in the past couple thousand years, however they should provide a rough estimate of dates.)
