Issues with driving LEDs with a CC driver
My circuit consists of 16 MP-3030-1100-40-90 connected in series which I'm driving with this CC driver @ Vs = 48V.


The LED datasheet says Vf can be up to 3.4 V at 150 mA so that's 16 x 3.4 = 54.4 V so your converter hasn't enough voltage headroom.


  
  Is the overheating caused by a lack of cooling? Are manufacturer's specs measured with in some kind of active cooling setup?
  


Yes. At maximum Vf, 3.4 V x 150 mA gives 0.51 W per LED to dissipate. That's easily doable with enough spacing and enough copper.


  The PCB they're mounted on is 1.6mm FR4 with minimal copper coverage.


That's your problem. You should design for minimum copper removal. You're paying for the copper anyway.


  
  It was my understanding that a CC driver keeps the current constant and the dimming happens by changing the voltage supplied to the LEDs. 
  


Almost. The constant current driver just controls the current. The voltage across the output terminals is just whatever the sum of the LED forward voltages works out to be. A CC (constant current) driver is complimentary to a CV (constant voltage) driver. The CV supplies whatever current is required at that voltage. The CC supplies whatever voltage is required to drive the required current.


  That's not what I'm seeing. Why?


It's not clear from your question what you are seeing.


  
  My plan for driving the LEDs was to use an accordingly sized CC driver. Since I can't seem to be able to drive these LEDs safely at If > 50mA, how do I prevent them from going over that number in hardware?
  




Figure 1. Output Current Adjustment by Variable Resistor. Source: LDU08-48.

See the datasheet. You can use a fixed or trimmer potentiometer.


  However, the maximum current I'm seeing after connecting this resistor in series is 31.3 mA. What gives?



Your resistor calculation looks OK. 
Measure the CC output voltage and double check that the voltage is still 48 V on load.
Use the built-in current control feature instead.
Short out half of the LEDs to see if the system works with a lower overall voltage requirement.