Well, aside from being very uncomfortable or irritating to put in and taking it out (both of which would probably trigger gag or sneezing reflexes), you'd negate many of the benefits that mucus provides.
Mucus acts as a filter when you breathe through your nose. Not only does it serve as scent-glue, but it can eliminate particles from entering your lungs - probably a bad place for things like coal-ash and stuff coming out of the tailpipes of cars. Particularly since air quality is so much worse after plants have stopped respirating. Keep in mind that mucus also catches bacteria and virii that would go straight into your lungs with a tube in place!
It also helps regulate both moisture and temperature levels. You may rue a stuffy nose on a cold winter day, but imagine being forced to inhale dry, freezing cold air and having your entire throat getting hoarse and spreading to the rest of your head (since sinuses reach further into your skull than your nose and throat).
So, while it would certainly relieve sinus pressure and allow someone to breathe through their nose after some adjustments, you might end up being sick more often, significantly colder if you were to go outside for any length of time, and your lungs would have more pollution in them.
Now, since I'm not a medical doctor, if you're still curious you could e-mail an Otolaryngologist (ENT or "Ear, Nose, Throat" Doctor) and get a much better picture.