まだ泣いてなくても、「don't look like you're going to cry 」より「don't cry」の方が英語で自然です。「泣き始めるのはやめよう」という意味で。
泣きそうな顔=look like (he's) going to cry; look as if (he's) going to cry
泣いた顔=look like (he's) been crying; look as if (he's) been crying
笑いそうな顔= look like/as if he's about to laugh/smile
笑った顔= look like he's been laughing (?)
暑そうな顔= look hot
寒そうな顔= look cold
怖そうな顔= look scared
眠そうな顔= look sleepy
「look」のかわりに時々「make a face」も言いますが、「look」の方が多いと思います。
「look」を使うと、顔だけじゃなくて身体中のことです。
You look hot.
He was looking as if he was going to cry.
He really looked scared.