As mentioned above, the fact that there have been successful productions of The Christmas Carol set in more modern times is proof that the novella is essentially timeless in its themes of greed and heartlessness. Nevertheless, the setting of the contemporary society of Dickens is of much significance to his times. For, as a social reformer, Dickens uses his text to expose the plight of the poor and the cruelty of the Poor Laws of 1834. Under these laws parishes were grouped into unions, and each union was compelled to build a workhouse. Inside these workhouses, conditions were extremely harsh, so only the most desperate asked to be in these workhouses. Families were separated and housed in different units. Much like prisoners, the poor were made to wear uniforms, sometimes having their heads shaved, and their food was inadequate. With strict regulations placed upon them, men and women,young and old, were made to work, often performing grueling tasks such as breaking rocks or picking oakum. Moreover, the children were often hired out to work in the mines or in factories, as Dickens himself was.
So, when Scrooge is approached by the two men who tell him "a few of us are endeavouring to raise a fund to buy the Poor some meat and drinks, and means of warmth" his contemporary readers would understand the significance of this remark and Scrooge's cruel reply,
"If they would rather die,...they had better do it and decrease the surplus population."
Other timely details of the narrative such as those in Stave Four in which the rag pickers take things from the lonely corpse of Scrooge and stand over him joking about the old miser would not be likely in modern times since bodies are taken to funeral homes and the like. Likewise, the plight of the Crachit family would not be such as it is with contemporary social programs that would provide medical care for Tiny Tim and aid the family in other ways.
There isn't really a specific year that A Christmas Carol is set in. When talking about the setting though, one can assume that it takes place sometime in the industrial age (so around mid-1800s) in London. I think the plot could work in just about any time and place, and we see this just about every Christmas since there are many productions on television parodying the book. The main ideas of the dangers of greed and the importance of family could be transferred into most any location and age.
However, the setting is important though and provides a powerful effect to the book. The main character is Scrooge, a wealthy man who symbolizes greed. This character of extreme wealth is really highlighted and emphasized in a backdrop of a poor London. Surely there are poor people everywhere, but having a setting of industrial London really allows the contrast between the selfish and rich Scrooge, compared to the overworked, meager, but kind Bob Cratchit.