In Caitanya-caritamrita (Antya 6.220) it is stated,

mahaprabhura bhakta-ganera vairagya pradhana

yaha dekhi' prita hana gaura-bhagavan

"Renunciation is the basic principle sustaining the lives of Sri Caitanya Mahaprabhu's devotees. Seeing this renunciation, Sri Caitanya Mahaprabhu, the Supreme Personality of Godhead, is extremely satisfied."

Because of false ego one considers oneself to be the proprietor and enjoyer of one's activities. Actually, Lord Krishna, the Personality of Godhead, is the proprietor and enjoyer of our activities; recognition of this fact in Krishna consciousness leads one to real renunciation. Every human being should perform his prescribed duty as an offering to the Supreme Lord. Then there will be no possibility of material entanglement. Lord Krishna clearly explains in Bhagavad-gita that prescribed duties performed as an offering to the Lord award liberation from material bondage. Sinful activities cannot be offered to the Lord but must be given up altogether. In effect, the distinction between piety and sin is made so that the living entities will become pious and eligible to surrender to the Supreme Lord. As explained in Bhagavad-gita (7.28),

yesham tv anta-gatam papam

jananam punya-karmanam

te dvandva-moha-nirmukta

bhajante mam dridha-vratah

"Persons who have acted piously in previous lives and in this life, whose sinful actions are completely eradicated and who are freed from the duality of delusion, engage themselves in My service with determination."

By thorough piety one's life becomes auspicious and freed from lamentation, illusion and fear, and one can then take to the path of Krishna consciousness.