DRG AN ENG RUN-UP BEFORE A SIMULATED INSTRUMENT FLT, THE CREW NOTICED THE CARB HEAT CONTROL WOULD ONLY MOVE ABOUT 1/2 OF ITS NORMAL TRAVEL. THE PLTS DISCUSSED THE MALFUNCTION & RAN THE ENG FOR ANOTHER 1 TO 2 MIN UNTIL THE OIL TEMP GAUGE INDCD A TEMP RISE. DRG TAKEOFF, THE ACFT WAS CLIMBING THRU ABOUT 400 FT AGL WHEN THE ENG BEGAN TO LOSE POWER. EMERGENCY PROCEDURES WERE FLWD, BUT THERE WAS NO SIGNIFICANT GAIN IN ENG POWER. AN ATTEMPT WAS MADE TO RETURN & LAND; HOWEVER, THE PLTS WERE UNABLE TO REACH THE ARPT. SUBSEQUENTLY, THE ACFT HIT TREETOPS & CRASHED SHORT OF THE RWY. LATER, AN EXAM REVEALED THERE WAS FROZEN WATER ON THE INSIDE OF THE ENG AIR INTAKE FILTER. NO PREIMPACT MECHANICAL MALFUNCTION WAS FOUND, EXCEPT FOR THE RESTRICTED TRAVEL OF THE CARB HEAT CONTROL. METEOROLOGICAL CONDS WERE RPRTD TO BE CONDUCIVE TO CARB ICING.