Keeping your dog safe from chewing up ornaments or electric cords is of paramount importance. Putting your tree up on a small table is one thing you might want to consider.
Tying up all electric cords out of reach of your dog is also helpful.
Putting packages on a table top instead of under the tree is fine and safer for your dog (and for the gifts).
Do not allow Holiday visitors to feed your dog extra treats or to feed your dog from the table. It is probably best to crate your dog during the meal and during the time that you open your gifts anyhow. And especially remember that cooked bones, especially turkey or chicken bones are highly dangerous for the dog!
Be sure your dog, especially if it is a puppy, gets his rest time undisturbed. Dogs stress out much more easily than you realize, and changes in schedule do disrupt their internal organs and digestive systems.
(This is where having a crate trained dog is especially important!)
If you are going to go away for the Holidays, get a trusted friend to check on your dogs in your own home or leave them at a reputable boarding kennel. Do not leave the care of your dogs to a ten year old child, for example, and do not just go away and leave them unattended. Even if you are only planning on being gone for the day, it is wise to have a stand by assistant in case of an emergency, especially since winter weather can be so unpredictable.
If you travel with your dogs and take them with you, then be sure that when you arrive at your destination, you are a responsible pet owner! Clean up after your dog, crate your dog when necessary, and be careful about introducing him to strange dogs that may live at the residence you are visiting. Some dogs are pretty touchy about strangers invading their territory and may be especially upset during a busy Holiday season when the household is in an uproar.