We'll start with the Tertullian argument. Tertullian was a church father from the third century who is mentioned by many Christmas detractors who think that he condemned Christmas decorations. ReligiousTolerance.org tells us the following
Tertullian (circa 160 - 230), an early Christian leader and a prolific writer, complained that too many fellow-Christians had copied the Pagan practice of adorning their houses with lamps and with wreathes of laurel at Christmas time.
The researchers at ReligiousTolerance.org clearly did not read the entire work when they looked this up. You can read all of Tertullian's work, On Idolatry, here.
In their strange attack on the Christmas Tree, they also tell us that pagans would never cut down a tree for decoration:
In Europe, Pagans in the past did not cut down evergreen trees, bring them into their homes and decorate them. That would have been far too destructive of nature.
So this practice, according to ReligiousTolerance.org, could not have come from pagan rites. But, again, they are making mistakes. The modern notion that ancient paganism is akin to our new-age earth worship religions is not supported by historical documentation. Pagan shrines most certainly made use of wood and other natural resources to build their many temples. Furthermore, there is no solidarity amongst pagan religions. There are worshipers of Pan, Isis, Thor, Roman Emperors, etc., and the term "paganism" does not refer to one set of beliefs. Anyone claiming that pagans all believed or acted in the same way has obviously not done their research. (Imagine, for example, claiming that all monotheists exclude shellfish from their diet. That is the logical fallacy that is being made.)
Then, they appeal to an old Egyptian practice:
Not having evergreen trees, the ancient Egyptians considered the palm tree to symbolize resurrection. They decorated their homes with its branches during the winter solstice.
(They do not quote a primary source for this observation, so who knows if it's even true.) Someone told me this, last year, when trying to convince me that Christmas was a pagan thing, and I was confused. What did this have to do with my Christmas tree? Did the person who made this argument think that Christmas trees symbolized the Resurrection for believers? Frankly, Christmas trees are not symbolic of anything to Christians - it's just a fun tradition. And it's hard to imagine the Egyptians lamenting their own lack of evergreen trees and using palm trees as the next best thing. Also, this makes no mention of calendar dates, so it probably had nothing to do with any holiday. This isn't related to Christmas.
Saturn must be invoked in each of these conversations. ReligiousTolerance.org reminds us of this:
The ancient Pagan Romans decorated their "trees with bits of metal and replicas of their god, Bacchus [a fertility god]. They also placed 12 candles on the tree in honor of their sun god" Their mid-winter festival of Saturnalia started on DEC-17 and often lasted until a few days after the Solstice.
Many people will tell you that Christians adapted the celebration of Saturn into their own holiday and called it "Christmas." This is the work of armchair historians, and not actual scholarship. If this were true, Christians would have celebrated Christmas sometimes before December 23, not after. Other than evergreens (which weren't associated with Christmas until much later), there's nothing that these two holidays have in common; they merely took place in the same month. There are no documents, or other evidence, that indicate that the church relied on pagan festivals for its own calendar, despite what we are often told. (And, ancient Hebrew writers claim that this festival came from their own Jewish holidays, which would explain the candles.)
What we have here is a confirmation bias. You can pick any date on the calendar and, with enough work, find out that an older culture celebrated a particular holiday on that day. It takes more than a correlation to prove something.
Finally, they try, clumsily, to quote the Bible in their strange discussion of the Christmas tree. If you work for ReligiousTolernace.org, then the following passage might seem to reference a modern holiday tradition to you:
Jeremiah 10:2-4: "Thus saith the LORD, Learn not the way of the heathen, and be not dismayed at the signs of heaven; for the heathen are dismayed at them. For the customs of the people are vain: for one cutteth a tree out of the forest, the work of the hands of the workman, with the axe. They deck it with silver and with gold; they fasten it with nails and with hammers, that it move not." (King James Version)
It's hard to say what they are trying to do with this quote. For starters, this is obviously not a reference to a Christmas tree. Unless you came from a pretty strange family, you probably didn't nail your Christmas tree to a wall to keep it from moving. And sliver and gold adornment are used for a lot of things, in this case these colors indicate that the thing being decorated is important. It's obviously a shrine that is being built. Christmas trees didn't exist until the late middle ages, so this must be a reference to something else.
There you have it; those are the arguments that were supposed to convince me that Christmas is not an original holiday. I actually have no idea what that ReligiousTolerance.org webpage is supposed to be arguing. It gives arguments for a non-Christian origin of Christmas decorations, but also states that these decorations are not from pagan sources. I'm not sure what they are up to, but their sense of history is very bad.
But we can soundly say that Christmas is not a pagan holiday. It is the Church's celebration of Jesus' birth, and there is no evidence to the contrary. Those who wish to hate the church will continue to launch these assaults, but history speaks of the truth.