What is the ruling on saying "By God, I beg you" (بالله عليك) and "I implore you" (أمانة عليك), and does it entail the ruling of an oath, obligating the action or atonement in case of non-fulfillment?

The statement "بالله عليك" (I adjure you by Allah) is considered an oath by Allah. Swearing by Allah is permissible if it pertains to a lawful matter. If the one who swears this oath breaks it, expiation becomes obligatory upon them. If a person swears you to something with this phrase, they break their oath if you do not do what they swore you to.

However, if the speaker means "أسألك بالله" (I ask you by Allah) by it, then it is not an oath but a request. Asking by Allah is Islamically prohibited, while responding to the one who asks by Allah is commendable, and some say it is obligatory.

As for the statement "أمانة عليك" (I entrust you with this/it's a trust upon you), it is not an oath and nothing results from it. It is not permissible if the intent behind it is to swear an oath, because swearing by other than Allah is impermissible.