The clear call coming from fuelwood researchers at SINTEF. The needles on the branches may cause problems because they contain substances such as terpenes that have a very high energy content. If you fill your stove with conifer branches they will generate enormous amounts of heat which in the worst case may lead to a chimney fire. The needles also contain a lot of nitrogen that can generate toxic gases such as hydrogen cyanide.
You can, of course, burn the trunk of the tree if you cut the branches off. If the trunk is still moist, either because the tree was recently cut or watered during the holidays, you should first let it dry out before you burn it.