'입이 무거워요' is used when you describe someone who keeps a secret well.
'입' is 'mouth' '무거워요' is' heavy'
'무거워요' is a present simple form of '무겁다'(dictionary form) -'ㅂ'irregular.
So '입이 무거워요' means 'mouth is heavy' literally
you can guess the meaning easily by imagining: someone has a heavy mouth so he or she doesn't open mouth easily and keeps secret.
A: 저 사실..... 비밀이 있어요. (Actually, I have a secret)
B: 뭐예요? 무슨 일이에요? (Really? what's that?)
A: 말해도 될까 모르겠어요. (I just wonder if it's ok that I share the secret with you)
B: 괜찮아요. 저는 입이 무거워요. 아무한테 말 안 할 거예요 ( Don't worry. I can zip my mouth.)
A: 진짜요? 사실은 저는, 저는, 아미예요 (Really? actually. I am ARMY(big fan of BTS)
B: 하하하! 사실 저도 아미예요. (haha. So am I)