It's an interesting point, and I'm in two minds about it.
Dickens was certainly a practising Christian, but it's evident from many of the portrayals of clergymen in his books that he thought that there was a lot of hypocrisy in the Church.
On the whole, I'd agree with those who say no, it's not overtly Christian. It's about treating people like human beings, no matter how poor they are, and how low their social status might be (Dickens was very big on social reform, of course).
I've seen it suggested in commentaries on ACC that Bob Cratchett's family celebration should be interpreted as a reinterpretation of the "Last Supper", with Tiny Tim in the role of Jesus, but I don't see this myself.
At the end of the day I see it as a great "feel-good" story, which tries to show that it's never too late to repent and start doing good.