The face you make when you're letting one go. The aligning of certain contours of your face is thought to have an impact on your ability to release anal gases.
Honey, you look funny when you're making your fart face.
A person who's face reminds you of a fart. Usually, they have round features, like round cheeks, round chin, round nose, plump lips etc and thick facial skin. Typically they have shiny, "dewy" (careful not to confuse with greasy) skin but in a gross way and it's like a sweaty butt. When you see them, you imagine their breath smelling like boiled eggs and a fresh can of roasted peanuts.
Person #1: Jennifer Lawrence is so HOT!
Person #2: Nah, she has a mega fart face. Look at her round face and round cheeks, looks like a rank fart.
Person #1: True
So there was these two blokes standing on the elevator back in the early 1900's, one man let out a particularly crotch staining fart while the other bloke just looked at him with this weirdfucking look of disgust on his face, and ever since then it's been known as fart face.