To help answer this question, I graphed average points scored by the week number in the NFL, based on every season from 1983 to 2009. The following graph shows the results.
As you can see, the line goes up and down a lot. The thin black line is a least-squared best fit line, which shows a general trend upward. So as the seasons progresses, and temperatures drop, average points scored increases slightly, but that could easily be random variation.
That is about as far as I can take it. For a general opinion about how weather affects sports betting in general, I turned to my friend Jason Been, who is an expert on the topic. Here is what he said:
In most cases, wind is the predominant aspect of weather that most affects a game; but, it is not the only one. In baseball and other outdoor sports, shadows can have an equal affect, especially during early- and late-season baseball afternoon games. Rain or snow isn’t as big of a factor as most people think in football, as it generally affects both the offense and defense equally. An example would be a defensive back against a wide receiver. The rain and snow would slow them down equally, thus giving neither side an edge. Wind can simply eliminate a passing game in football along with kicking. I have seen games where a passing team has been forced to run the ball almost every play due to a strong crosswind. It doesn’t happen often, but on occasion the wind is the ultimate decider of a game.
This question was raised and discussed in the forum of my companion site Wizard of Vegas.