You will lose.
As the Offering is resolving you follow the instructions in order, thus first you destroy the Lich, then you gain life. Since Lich is no longer on the battlefield its replacement effect doesn't apply so you will not draw cards instead of gaining life. Then once the Offering is done resolving the Lich's leaves the battlefield trigger will go on the stack and you will lose the game when it resolves.
608.2c The controller of the spell or ability follows its instructions in the order written. However, replacement effects may modify these actions.