For me, it looks more like he has big red eyes.. Maybe because he's mad because he's dying? But whatever, that red hue on Mario's face is really questioning, but today everyone seems to accept it, plus I find Mario strange without that red hue:
It does look like Mario's shouting/screaming out of anger. I found it a bit strange or creepy many years later. I don't watch anime anymore, so I don't know if it's some kind of anger.
Luigi's palette has been changed, because huge green eyes and a green mouth look weird.
As for Mario, I guess it has something to do with anger of annoyance. Characters often turn red in cartoons if they're mad/annoyed. I think Nintendo had the same idea, however they couldn't change Luigi's palettes probably because of SNES palette limitations or the fact, it would force them to change a lot of palettes and mess with them to keep everything fine, without strange palettes.
For me it's most likely Mario's blushing out of embarassment, as if the game is telling you "Aw, you died! What a shame!". Luigi's face would be consequence for sharing the same GFX as Mario. Considering it happens in only one frame, Nintendo thought it was minor enough to leave it as is. We all know Nintendo didn't mind ignoring minor issues in SMW back then.
Edit: it probably happens when the princess kisses Luigi too, but still minor enough for them, I guess.
I don't think his face is showing any sign of anger nor it's the hat's shadow, as it would have to be way smaller than that.
I'll be honest, I never liked that face. It actually gave me nightmares when I was a little kid when I first played the game. I'm obviously not scared of it anymore but it still makes me uncomfortable... eh.