Yes, the circuit will not work with the TL082, because its inputs need to be a couple of volts from the TL082's negative rail for it to work. A single supply op-amp, such as the LM358 is required for it to work.
I don't understand why the circuit you posted wouldn't work in RL though. The inputs of the opamps are nowhere near -15V at any time, right?
Also, the simulation I did with the TL082 seems to suggest that it would work simply by adding a coupling cap at the OP output. What am I missing here please?
Please disregard my previous post. It was about the temperature indicator circuit, not the circuit I posted.
The circuit I posted will not work because the two op-amps will not have exactly the same gain and offset voltages, as is the case with component models. U1 is operating as an inverting amplifier, with a gain of 10 and as you can see, is working perfectly in your circuit. The problem is U2 which has no negative feedback and is operating open-loop. In your circuit, the TL082 model must work slightly differently, when the inputs are connected the opposite way, presumably because of its offset voltage, which is taking the output more positive, but it still has the same gain, so the signal is the same amplitude. If you built this in real life, expect this to be worse. U2's output will most likely saturate at either supply rail. It might also output a square wave or a very tiny signal biased somewhere in between the power rails. The behaviour will be unpredictable, because the open loop gain and the offset voltage of U2 is unknown.