What is the ruling on messages that include ذكر (remembrance of God) while obligating the recipient to resend them to others, claiming it is an "amanah (trust) on your neck until the Day of Judgment"?

You are not obligated to send this to others, especially if it involves a financial cost. The sender's statement, "It is a trust upon your neck," is incorrect because you have not accepted this trust nor promised to do so.