What is the ruling on saying "by your conscience/oath," and does it necessitate an expiation for lying about it, and what is the ruling on lying after someone makes an oath by saying, "By God Almighty, you must tell me the truth"?

The statement by the one who says, "by your conscience," does not constitute an oath that necessitates expiation, because a binding oath must be by one of the names of Allah the Almighty or one of His attributes. As for lying, it is forbidden and is only permissible if it becomes a means to avert harm. A Muslim must be truthful in their words and actions, especially if truthfulness is requested of them, and they should beware of lying, which is reprehensible both legally and by nature.