When everything seems to be setup right and yet doesn't work, you first troubleshooting step should be to carefully check your wiring, make sure you are counting ic pins in the right direction, so look carefully for wire shorts, even a wrong concept of circuit operation.
Read the specsheets to make sure you are identifying and connecting pins correctly.
Step2: Get a scope or DVM and trace the power input to output and see which point shows unexpected readings.
With just a quick glance at your design, your circuit looks to be a current/voltage source rather than a current sink.
Why so complicated? Why use the LT3080 for this purpose??
With just one transistor, or one power darlington, one LED and two resistors, one variable, you can make a fairly good current sink.
Use an op-amp to measure the current through a shunt resistor to control the current through a power pass transistor to regulate current. Google current regulators.
The circuit must follow the laws of physics! If it doesn't, you've made some mistake in expected circuit operation, wiring or component values, etc.
Fascinating, you don't describe the current range you want to control..nor the voltage range of compliance.
Something about your goals in creating a current sink is important to reveal to help anyone possibly understand your circuit.
Could you possibly explain how your circuit works to another person?