Friar Lawrence is only commenting that Romeo has been shedding many salty tears over Rosaline and now suddenly has forgotten her and fallen in love with Juliet. Brine is salty, like tears. Sallow is a sickly yellow color, signifying in this context that Romeo's complexion looks pale and sickly because he hasn't been able to eat due to pining over Rosaline, and also that his cheeks have lost their natural healthy color because they have been washed by so many briny tears for Rosaline.
I love Friar Lawrence. He has his faults for sure, but I love the fact that he is willing to call Romeo out on his ridiculous behavior. The line of dialogue in question occurs in Act 2, Scene 3. Romeo has just come to visit the Friar after having fallen madly in love with dear Juliet. He's completely enamored with her, and Romeo wants to know if the friar will agree to marry the young couple.
"but this I pray,
That thou consent to marry us to-day."
Romeo's request strikes the friar as odd, because it was only a bit earlier that Romeo came to the friar to mope, moan, and complain that the beautiful Rosaline didn't reciprocate his feelings of love for her. Friar Lawrence is simply commenting on the fact that Romeo was able to wash away his salty tears and sickly skin rather quickly for somebody that was supposedly so in love earlier. I completely agree with the friar, and it's Romeo's quick to fall in and out of love personality that has caused me to never like Romeo.