Potential divider to check IV characteristics
If I've understood your description these are the three circuits above.

Start with circuit 3. 

If the variable resistor is set to maximum then current will flow throw the diode. You won't be able to get any values of current below this so some of the IV characteristic cannot be measured. 

If the variable resistor is set to zero then the diode forms a short circuit for the power supply. There is nothing to limit the current so it goes 'bang' and releases the magic blue smoke.

Circuit 2

This works fine (you can get the IV range down to zero current). However, if you get the wiper near to the top of the potential divider there is nothing to limit the current through the diode. Once again 'bang' and a puff of blue smoke.

Circuit 1. 

This works fine at all settings of the potential divider.