Hmmm, you might want to check the pinout. From Gate to either Source or Drain, you should measure wide open. I get the feeling you are measuring Source to Drain. .. or you're measuring your body resistance by holding onto the probes. There's a back diode across the channel, built in to this particular MOSFET.. Here's a link to a Fairchild version:
http://www.pcpaudio.com/pcpfiles/transistores/IRFP240.pdf The Fairchild, the Chinese, and the International Rectifier version should all follow the same pinout.
Also with anything "MOS", be sure you follow safe ESD practices. While these devices have very rugged specs while being soldered into a circuit, static electricity applied to the Gate can cause punch through and cause a leak or a short to the channel.
Using the datasheet, and using the meter's diode test, put the meter's red lead to the Cathode (K) of the internal zener diode and the meter's black lead to the Anode (A) of the internal zener diode. Then touch the gate lead, with a jumper, to the red lead.... measure that. Remove the jumper from the red lead. Now attach the Gate, with a jumper, to the black lead. Measure that. Post those results. If you're getting no difference, it is possible/likely that the meter's diode test does not force enough voltage to turn on the gate. The force voltage will be under the MOSFET's V
GS threshold voltage. getting around that is more complicated and requires something to increase the Gate voltage, either through the use of an external battery or power supply.
Sorry it's not that easy to explain, but FET's especially MOSFET's have different flavors and transistor tests can't be directly applied in many/most cases.