When you get the hiccups, the diaphragm contracts in an involuntary manner, this is known as a spasm. As it contracts, the glottis (commonly known as the vocal cords) at the back of the throat snaps shut.
The effect of this sudden action on air flow through your nose and mouth is twofold. Firstly, blood pools in your pharynx (throat) because it can't escape quickly enough. This increases your risk of suffocation since less oxygen reaches your brain and other organs. Secondly, since the glottis shuts off airflow to such a large area of your mouth, acidity (particularly from stomach acids) builds up very rapidly in that area. This combination can cause intense pain, called hiccup migraine, and even teary eyes - because all of that acidic secretion gets blown into your face!