In the expression the hell out of, "the hell" (lower case) is used to emphasize the action you are referring to:
- Used in verbal phrases to emphasize force, speed, etc.
The hell has a long history in idiomatic expressions where its function is mainly to add emphasis to the literal meaning of the sentence. According to Google Books the expression "beat the hell out of" is from 1920s and precedes other similar constructions such as "beat the crap/shit out of".
In the following piece, Professor, and ELU user John Lawler comments that:
To start with, "the hell" must be distinguished from "Hell!", or "Oh, hell!", which are full utterances (traditionally, "interjections", the last and least of the classical Eight Parts of Speech), the sort of thing you say when you've made a mistake, had a mistake made for you, or otherwise experienced the displeasure of Fortune. They are linguistically unusual in having no syntax -- "Oh, hell!" is a full utterance (though hardly a 'sentence' - no subject, no verb, etc.) and needs no further complement.
There are also some fixed phrases. "What the hell" is used to express disregard for conventional procedure and precautions, in varying degrees (the phrase has some of the same implications as "devil-may-care"). It is usually a phrasal interjection, without further syntactic ramifications: