I have sometimes heard that killing someone frees that person of their sins and that the killer takes on their sins. Is this true or is there any reference to this in any Bible verse?
This is provably false. Sins are not inheritable nor can they be transferred to another person. "The soul who sins shall die. The son shall not bear the guilt of the father, nor the father bear the guilt of the son. The righteousness of the righteous shall be upon himself, and the wickedness of the wicked shall be upon himself" (Ezekiel 18:20). A person is judged by his own actions and not those of another person. "But in accordance with your hardness and your impenitent heart you are treasuring up for yourself wrath in the day of wrath and revelation of the righteous judgment of God, who "will render to each one according to his deeds"" (Romans 2:5-6).
Murder is a sin that does not benefit its victims.
"Therefore, my beloved, as you have always obeyed, not as in my presence only, but now much more in my absence, work out your own salvation with fear and trembling; for it is God who works in you both to will and to do for His good pleasure" (Philippians 2:12-13).