Is it obligatory to fulfill the request of someone who says, "For God's sake," and does not fulfilling it incur sin or expiation?

If the utterer's intention behind the phrase "I adjure you by Allah" (أقسم بالله عليك) is a binding oath, then fulfilling it is sunnah (recommended), unless it involves a sin or leads to harm. It is not obligatory, and there is no sin upon the one who does not fulfill the request. However, if the intention is "I ask you by Allah" (أسألك بالله), then asking by Allah is forbidden by sharee'ah (Islamic law). The preponderant view is that answering such a request is obligatory, unless it involves a sin or leads to harm.