Tel Aviv and Jerusalem are as safe on these days as on all other days. You should differentiate between the two cities, there is a difference in how these days are celebrated in each:
Tel Aviv: Christmas is pretty much nonexistent in most of Tel Aviv, there is a small number of public/touristic events related to the holiday, but they are all happening in Jaffo.
Jerusalem: Christmas is (not surprisingly) more significant in Jerusalem and there are more events celebrating and relating to the holiday. But again, they are localized to specific areas and not citywide, like you might find in European and other places (more on this later).
Because Christmas is a Christian holiday, and the more "explosive" times are around Jewish or Muslim holidays, it should be as safe as every normal day, if not safer due to the mutual interest of not harming tourists and visitors.
The new years eve is even less of an issue, it's not really a special day for most people here, and as such there are no public events for it, there are some clubs, restaurants, bars, etc. that will have special events for the day, but they are all inside the relevant establishment, and not public. There should be no added risk on that day.
Few more general notes and recommendations:
While I can't tell from personal experience, but the general consensus is that the best place to experience Christmas in Israel is in Nazareth, if you want to experience a special Christmas in the Holy Land this would probably be the best place for it. The other place is Bethlehem, in the Palestinian territories. The place where Jesus was born is probably the best place to celebrate his birth.
Pickpocketing is not really a problem in Israel as it is other popular attractions in the world, so at least here you can walk a little easier on the street (but still be careful and use common sense).
The old city of Jerusalem is really filled with lots of armed personnel for protection (and is also lined with CCTV cameras filming every corner), but they are not from the military, they are policemen and women from the Border Police which is the Israeli gendarmerie. The main difference for a visitor is that they are trained to deal with civilians and the public and, I believe, that once you understand that they are police officers and not soldiers, it becomes less threatening to a visitor.