Is there a variable forward voltage diode-like device or circuit construction?
simulate this circuit &ndash; Schematic created using CircuitLab

Something like this?

It's not a single device, but it is simple(ish).

Heatsink Q1 for the LED current dissipation.

Choose R1/R2 to give you a suitable ratio between the 'diode' voltage and your control voltage, the values shown give you 3:1.

D1 is to prevent reverse breakdown of Q1 BE junction, should you use a high voltage supply to the op-amp.

Use a darlington for Q1 if you want to sink a lot of current. Even with a darlington and D1, you ought to be able to get down to 2.2v if your opamp output goes down to near -ve rail.

Put a small resistor in series with the load if you want to increase the impedance to mimic a real LED load, this sink will be very low impedance.

Bonus component. Add R3 to alow you to monitor the current with a voltmeter. Obviously it doesn't measure the current through the other resistors (they could be increased somewhat), or in the base of the transistor (1-2% in a power PNP, &lt;0.1% in a power darlington), but it measures most of it with an accuracy comparable to your DMM and its own tolerance. If you do increase R1 and R2 above 100k or so, put a few 10s of pFs across R1, to ensure stability.