I dont think you could burn up those if you connected them directly across the 9V battery. Most of that type of headphone are ~35Ω dynamic.
The TL071 opamp is not capable of driving such a low-impedance load. It is spec'd for a >=2KΩ load impedance. If you are paralleling the two earbuds, you are asking it to drive ~16Ω, which ain't gonna happen...
You need a LM386 or similar audio amp to drive such a low load impedance.
If they overheated and burned up you MUST have connected them wrong (and probably directly across the battery? - which would put 2.5W though each headphone).
Assuming you connected them directly across the battery in series with the 100 ohm (no capacitor in series, or it was S/C), then that would limit the power to 0.6W, or 0.35W each if in parallel.
Even assuming the circuit was unstable and oscillating it couldn't produce enough power to kill the phones.
You just need to learn to read the data sheets for the ICs you include in your circuits. If you got the original circuit off the web, remember that most of the posted circuits are put there by people who don't know how to read data sheets.
You can always use the existing circuit as a preamp, and then add a LM386 stage to drive the headphones.
" If you got the original circuit off the web, remember that most of the posted circuits are put there by people who don't know how to read data sheets." That is MY CIRCUIT and notes. I read the datasheet many years ago.
The plug on the earbuds makes it impossible to connect them in-phase in series. If the signal feeds the tip and ring of the plug (then the sleeve is not connected) then the earbuds are in series but are out-of-phase which sounds and feels weird.
Most earbuds today are 32 ohms for each earphone. The little 10uF capacitor on the output feeds 20Hz to a load that is 700 ohms in series with the 100 ohms at the output which will overload the opamp.