You know that moment when you're eating an overly Sriracha-ed sandwich, and all of a sudden your nose is running more than a kindergartener during cold season? Turns out that that nose running induced by spicy foods is the foods' way of telling you not to eat them.
One main reason for all the snot is capsaicin, a chemical in many peppers that's also behind the burning sensation your tongue feels. Capsaicin is supposed to deter mammals from eating the seeds, so that the plant can reproduce better. It also keeps the plants safe from many kinds of fungus.
Another reason is allyl isothiocyanate (try saying that three times fast), which serves a similar function. You may be familiar with allyl isothiocyanate, especially if you're a big mustard or horseradish fan. This colorless oil also protects from fungi, causes your mouth to feel "hot", and is supposed to protect the plant from those who want to eat it. Of course, evolution didn't count on just how much we would love the burn of our hot sauces.
These chemicals also do quite a number on your mucus membranes. They inflame the tissue, which makes your body produce more mucus to try to get them the hell out of your system. Basically, the runny nose is just your body's way of trying to get rid of whatever it is you just ate.
There's a similar thing going on here with the way your eyes water when you eat spicy foods. The capsaicin and the allyl isothiocyanate will also irritate the membranes in your eyes, so that your tear ducts try to flush out the irritants. But that just makes your nose even runnier. Basically, if you're going to douse your food in hot sauce, you're pretty much just going to have to deal with getting a face full of snot.