Suppose a business in a state wants to run a contest for its customers. For this example, say it's a snowshoe race. However in order to make the competition more interesting, they want to limit the competitors to residents of warm states and exclude residents of states like Minnesota and New York.
Are there any legal problems with this?
There are numerous laws pertaining to protected classes, whereby it is prohibited to discriminate on the basis of that class for certain purposes. For example, it is generally illegal to consider race in public accommodation, employment, housing, or voting. Age is only a very limited a protected category. In addition, there are anti-discrimination laws at the federal, state and municipal level, where it may be municipally illegal to discriminate in terms of a certain class where there is no state of federal protection.
State of residence is not a federally protected class for any purpose, and I have not heard of any state or municipality having a law prohibiting discrimination (presumably for "public accommodation") on the basis of state of residence. So your problems in doing this would not be legal problems.
