1N4739A 9.1V Zener Diode Reverse Current Decreases with Temperature—Why?

I wanted to study the effect of temperature change on diode characteristics. I used a 1N4739A 9.1 V Zener diode for that and connected it in reverse bias. Since the meter is not that sensitive to measure leakage current, I increased the voltage above 9 volt to breakdown region. Introducing a heat source to the vicinity of the diode, the current starts decreasing. How do we explain that behaviour? The diode resistance is supposed to decrease with increase in temperature right?

The reverse breakdown voltage of a Zener diode usually increases with temperature. That's why a temperature-compensated Zener diode is a reverse-biassed Zener diode in series with a forward-biased ordinary silicon diode.
Decades ago, General Electric made reference amplifiers, where the base-emitter diode of a silicon diode temperature-compensated a zener diode in series with the emitter on the same chip. The base-to-Zener voltage was independent of temperature, so could be used as the reference for a voltage regulator.

I hope you are not just putting a power supply across a Zener diode?  A zener needs a resistor in series to limit the current to a safe value.  It dissipates power so for example a 5V, 1W Zener can have a maximum of 0.2A (200mA) flowing through it (5V x 0.2A = 1W.Zeners have temperature coefficients which vary with the Zener voltage. Assuming a constant current through the Zener, below 5.6V the voltage will decrease with temperature, above 5.6V it will increase with temp.  At about 5.6V it is pretty constant, which is why 5.6V Zeners were a favourite for reference voltages for power supplie.The Wikipedia articla has a good treatment of Zeners, under the heading "operation" is a good description of the temperature effects.

Thanks for the information. It was helpful.  I had used a 1 k resistor to limit the current. 

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Correction, "transistor" instead of "diode": Decades ago, General Electric made four-lead monolithic reference amplifiers, where the
base-emitter diode of a silicon transistor temperature-compensated a zener
diode in series with the emitter on the same chip. The base-to-Zener
voltage was independent of temperature, so could be used as the
reference for a voltage regulator.