According to this Wikipedia article, yes, some states do have such laws. (My answer is based on the sources listed in this article, so if some other state has a law that wasn't listed here, I wouldn't know about it.)
Arizona law says: "Except during the final year of the term being served, no incumbent of a salaried elective office, whether holding by election or appointment, may offer himself for nomination or election to any salaried local, state or federal office."
Florida law says: "No officer may qualify as a candidate for another state, district, county, or municipal public office if the terms or any part thereof run concurrently with each other without resigning from the office he or she presently holds."
According to the Georgia constitution, "The office of any state, county, or municipal elected official shall be declared vacant upon such elected official qualifying, in a general primary or general election, or special primary or special election, for another state, county, or municipal elective office or qualifying for the House of Representatives or the Senate of the United States if the term of the office for which such official is qualifying for begins more than 30 days prior to the expiration of such official's present term of office."
According to the Hawaii constitution: "Any elected public officer shall resign from that office before being eligible as a candidate for another public office, if the term of the office sought begins before the end of the term of the office held."
The Texas constitition also has a provision that running for another office constitutes an automatic resignation, although only certain offices are affected.