How do I interpret a regression model when there are impossible additive effects? Let's say I have a model of count data as a function of the month of the year along with an additive effect of season (factor with 2 levels Wet and Dry which correspond to Jan - June and July to Dec respectively).
count ~ month + season

My regression will give some main effect estimates for the two parameters. Let's say I have a positive effect of wet season. How do I interpret this when the effect is on months that are not in the wet season?

Turning my comment into an answer:
If season is just defined in terms of month and consists of the same set of months each year, there isn't much point in having both in the same model because of the covariance between the two. month alone will capture the season effect as well as more fine-grained variation. This might be less of a problem if the dry/rainy seasons occurred in different months each year, but in general this isn't likely to work well.
Now to restate your bolded question: "How do I interpret the positive effect of a wet season when the effect is on months that are not in the wet season?"
Even ignoring the model problems I discussed earlier, the question doesn't really make sense. The parameter value for season distinguishes between dry and wet seasons. Assuming 'dry' is your reference category, then the season parameter value applies only to 'wet' periods (and vice versa). There's no effect of wet season applied to months not in the wet season. The question What is a contrast matrix? might help make this clearer, and I also suggest looking up 'dummy coding' on this site.
