Below is a voltage regulator circuit for which I measured Vo and Io over a range of load resistance values. Why is there a slight drop in Vo when Io is increased, does it have something to do with the 1N5232B zener diode? (Vdo = 0.7 V, Vz = 5.6V).
That's behaving more like a 5.1 V Zener. At zero output current the voltage is a little over 5.1 V. At 3 mA it's about 5.1 V.
Once you draw more than 4 mA the voltage drop across the 1k resistor will be > 4 V so the output voltage will drop.
I recommend that you double-check the part number and specification. If you're sure it's a 4 V Zener (although your test measurements say that it's not) then add the part number and link to datasheet into your question.