Christmas trees are considered to be mildly toxic. The fir tree oils can irritating to the mouth and stomach, causing excessive drooling or vomiting. The tree needles are not easily digested either; possibly causing GI irritation, vomiting, gastrointestinal obstruction or puncture.
Even though the question refers to «live» trees, artificial trees, such as the one in the photo, are also dangerous when eaten. The principal things to worry about are toxin release from the artificial material and intestinal obstruction (not digestible).
As noted earlier, the amount of trouble depends on how much is consumed. Many times, pets don’t consume mass quantities of tree material.
I would recommend confining your pets away from the tree when you are not home.
This will allow you to be able to «supervise» any tree or plant eating activity. Other plants, such as Mistletoe and Holly are also poisonous.
What to Look For
If your pet has chewed on the Christmas tree or other plants, monitor for any changes of behavior (excessive licking, salivating), appetite, activity, water consumption, vomiting and diarrhea.
Additional Tree Safety Concerns
While we are on the subject of Christmas trees, also consider the tree water. Preservatives, pesticides, fertilizers and other agents, such as aspirin, are commonly used in the tree water to keep the tree fresh. These may have harmful or deadly consequences for cats and dogs (and children) who drink the water! A covered tree water dish is the safest.