According to this article, if the Republican Party chooses Kansas City for its 2016 convention site, the convention will start on June 27. If one of the other three cities that is under consideration is chosen, it will start on July 18. The city will be chosen in August 2014.
If the convention does start on June 27, that will be the earliest major party presidential convention since 1948, when Republicans met starting on June 21 in Philadelphia. A convention that early will put pressure on states that hold presidential primaries in June to move them to an earlier month. Those states are California, New Jersey, Montana, South Dakota, New Mexico, and Utah.