The answer surprisingly is yes. A real live Christmas tree is a great place for mold to grow, which can inflame nasal passages. The mold spores, which are invisible, like moist trees which have been cut for an extended period of time.
The best answer is to use the fake tree. Mold cannot grow on these trees. If you really want to use the live tree, do a few things. First make sure the tree was recently cut to avoid a buildup of mold. Second of all, keep the tree on the drier side as mold does not like drier environments. Finally, keep the tree for as short a period of time as possible.