New Hampshire and Illinois, for example, have "reverse evasion" statutes. These laws prevent non-residents from getting married if their home state would not recognize the marriage. They are not designed to specifically affect same-sex couples. These are laws that have been on the books for years.
The Illinois statute states: (750 ILCS 5/217) (from Ch. 40, par. 217)
Sec. 217. Marriage by Non-residents - When Void.) No marriage shall be contracted in this state by a party residing and intending to continue to reside in another state or jurisdiction if such marriage would be void if contracted in such other state or jurisdiction and every marriage celebrated in this state in violation of this provision shall be null and void. (Source: P.A. 80-923.)
Same-sex couples living in a non-recognition jurisdiction cannot enter into marriage in their home state. Since the marriage would be void in their home state, Illinois law prohibits the couple from marrying in that state.
The New Hampshire legislature is considering legislation that will repeal their reverse evasion statute but there is no similar action pending in Illinois.
There have been recent news reports from Illinois that many of the licenses being issued in the state are for lesbian and gay couples from Ohio and Indiana. It is important to find out how these statutes affect the validity of the marriage. If the marriage is not valid where celebrated, the federal government will not recognize it.
It may be necessary for the couple to marry in another marriage equality jurisdiction. In time, these statutes will not be as important because same-sex marriages will be recognized throughout the country.