Is it true that a man's funeral becomes heavier on the day he dies if he trimmed his chest hair?

This statement is incorrect and has no proof. Shaving chest hair is permissible as it is an act that the Shari'ah has remained silent about. A permissible act carries neither punishment nor reward, neither blame nor praise. Moreover, the state of the deceased is among the matters of the unseen, about which it is impermissible to speak without knowledge. The heaviness or lightness of a funeral bier is not an indicator of the deceased's righteousness or evil.