If you made a family event out of tree hunting at any one our local Christmas Tree Farms, your tree is much fresher and will easily last until New Year's, even if you put it up the first of the month. For other trees, the needles are likely already drying and the tree is no longer taking up any water. Those trees need to be watched carefully and any heat sources, like a lit candle, need to be kept far away for safety reasons. Older style incandescent string lights also can generate a lot of heat, so be careful of those as well.
Once the decision is made to take the tree down, you can recycle the entire tree by using cut up evergreen branches over your tender perennials in the garden or other perennials that are early spring targets for the rabbits. Spring bulbs can benefit from the protection, but not from the cold, from early rabbit feeding next spring too. Another use for those branches is decorating the outdoor pots by the front door after other holiday decorations are removed. You can use the leftover tree trunk to grow climbing peas or pole beans next spring since you cannot burn the trunk indoors in the fireplace because of all the sap. All those fallen needles under the tree in the house can be collected and added to the compost bin or pile, or scattered in the flower beds on the snow or bare ground.
If you want to leave the tree whole, tie it to another tree or other structure, and it will provide shelter for the birds, especially if you feed the birds in your yard. An interesting project for the kids over the holiday break and into January and February is to make edible decorations for the tree that benefit the birds. Pinecones covered with peanut butter, strings of popcorn and fresh cranberries are easy ones. Making suet balls to place in the tree is welcomed by the larger birds.
While not exactly related to the Christmas tree, your other holiday gift plants should be recycled as well via the compost pile or bin. While poinsettias can last many weeks indoors, most of them did not receive regular watering or enough light to continue to look good. Paper whites is another holiday plant that is not expected to be kept. Some that should be kept are amaryllis, Thanksgiving and Christmas Cacti, and cyclamen. The cacti can be grown outdoors in their pots for the summer and will re-bloom for us during the holiday season. Amaryllis can be planted directly in the ground and dug up before frost, trimming the tops away and stored until January or February. When repotted indoors and they will then re-bloom as well.