How did Father’s sweet, sing-song voice affect Jonas on their last evening together?

You are talking about the episode which happens in chapter 21 of The Giver, by Lois Lowry. Father has decided that he is going to "release" Gabriel, who is just a toddler, the following morning. Jonas, however, has previously learned (chapter 19) that "releasing" actually means that the child is going to be killed by a lethal injection and that his body is going to placed in a carton and disposed of.
Father looks at Gabriel and softly sings "It's bye-bye to you, Gabe, in the morning." As Jonas knows what is really going to happen, his father's naive and playful tone is especially disturbing and quite grotesque. Father appears, and actually is, completely devoid of emotion and conscience. Father's sing-song voice is the final straw for Jonas, the element that cements his decision to leave the following morning, bringing Gabriel along in order to save him. He knows that if he stays Gabriel, does not stand a chance, so his only way out is to escape and take the baby with him.