I just want to preface my answer by saying that your question is a source of confusion for two reasons: "elements" could refer to either film or literature and the words "Oliver Twist's story" make it questionable which movie version you are speaking about. I have a feeling that elements of film such as camera work, lighting, scenery, etc. aren't what you were getting at here. I also think you are most likely referring to the film Oliver! Therefore, I will answer the question about the film Oliver! and from a very general viewpoint.
I think someone could make a strong case that dialect, lyrics, and costume (among numerous other "elements") absolutely serve to highlight the social background of the characters in Oliver!
One of the absolute best examples of the dialect of the lower classes in London at the time is can be found in the song "Consider Yourself." Of course, this song also serves to provide examples of the lyrics as well. Take the simply states first line: "Consider yourself at home." Oh sure, it sounds regular enough. I'm tellin' ya', though, when the Artful Dodger says it, it's more like, "Consi'er yo'sel a' ome." Quite honestly, no member of the upper class (or even the middle class, mind you) would be caught dead talking like that. It puts a person directly into the gutter, ... similar to what happens to Eliza Doolittle in My Fair Lady (or Pygmalion).
Just to focus on lyrics for a second, one can't deny "Food, Glorious Food." Orphans wishing for a full meal, ... reminds me of the "empty-belly" life in Annie's "It's a Hard Knock Life." A commentary on social background? Take a gander:
Food, glorious food!
Don't care what it looks like --
Don't care what the cook's like.
Just thinking of growing fat --
Our senses go reeling
One moment of knowing that
No doubt about the social background when it comes to costumes, either. Oh, and who could forget little Oliver's dirty orphan costume when he asked, "Please sir, may I have some more?" Browns, grays, and blacks all mixed together obviously from different hand-me-down outfits from others. Saddest little bugger you've ever seen. And the Artful Dodger is even worse with his little nasty cravat and old hat that obviously used to be very fancy.
There are so many other elements that comment on social background in this film, I could have a field day. Hopefully this post will serve to get you started.