The private pilot reported that during a cross-country flight, the engine rpm decreased and he applied carburetor heat. He kept the heat on for a period, and then turned it off after the rpm increased to 2500. The rpm dropped immediately to 1000 rpm, and went to 2000 rpm upon the reapplication of carburetor heat, but then decayed again. The pilot made an emergency landing at an airport, but overran the end of the runway and collided with trees. An inspection of the airplane by an FAA inspector disclosed that the carburetor heat control was not functioning properly, and that the engine fuel mixture arm was slipping on the mixture control cable. In his written report to the NTSB, the pilot indicated there were no mechanical issues with the airplane he was aware of that precipitated the accident. A carburetor icing probability chart indicated that temperature and humidity conditions in the vicinity of the accident were conducive to carburetor icing.