Beating the chest is a way to express grief and sadness among millions of human beings in different parts of the world. A'ysha daughter of Abu Bakr did beat her chest when her father died. (Fath Al-Baari Fi Sharh Saheeh Al-Bukhari, Hadeeth 2420).
Lovers of Ahlul Bayt (AS) love Imam Husain (AS) more than their love to their own families and children, and feel very sad when they remember the tragedies which took place in Karbala and other places. That is why they beat their chest to express their noble feelings of sadness on Imam Husain (AS).
Islam gives everyone his full right to express his feelings in the way suitable to him as far as it does not include a sinful act. No one can drink alcohol to express his sadness on any matter because it is a sinful to drink alcohol. Beating chest is never a sinful, and it becomes a noble act when it is to express noble feelings for the Most Noble persons who are Ahlul Bayt (AS).
Wassalam.