auf die Tränendrüse drücken — To press on the tear gland. Duden marks it as mildly disparaging. It means trying to produce emotion and sentimentality through the way something is presented.
die Gefühlsduselei — Emotional fuzziness. Duden marks it as colloquial and disparaging: thinking or behavior driven by exaggerated feeling.
rührselig — Overly sentimental, too easily moved to tears.
die Mitleidstour, die Mitleidsmasche — The pity routine, the pity trick. Tour and Masche both name a method used to get something. Duden gives the example die Tour zieht bei mir nicht! That routine does not work on me.