Did he chew it up or swallow it whole?
How long ago did he eat this?
Based on the size of the dog and the pine cone it is unlikely this will cause any problems. To much time has passed to have him vomit this back up.
I am concerned that bits and pieces can lodge in your dogs digestive track and cause a blockage. You will need to give him metmucil, the usual dose is 1 to 5 g per pound (2 to 10 g//kg) every 12 to 24 hours. This will help encase the cone and cause less damage. Put it on top of a food he will normally eat. While he may be able to pass it on his own this is not a guarantee. If you have no vet in your are open today feed him as much breads as you can to help encase the cone as well.You will need to keep a close eye on his stools for the next few days to make sure this passes as well. You will want to watch him for signs of a blockage as well. When a pet has a blockage, they may exhibit any of the following symptoms: vomiting, loss of appetite, distended belly, an absence of feces or a change in the feces, blood in the stool or vomit, drooling, and retching or trying to vomit with the meal coming up or nothing coming up. If your pet exhibits one or more of these signs a trip to the vet is highly recommended.