Yeah, forget about getting any help on this until you post a photo of your circuit, a schematic that shows anything that is not in the schematic you linked of the op amp circuit and your code.
At this point it is anyone's guess as to whether or not it is hardware or software. We need the EXACT sketch that your running when this occurs, and a schematic of the entire setup when this occurs.
FYI. - Try isolating the problem by separating the led power from the ATtiny power. Also separate the code . Run just the op amp
code with everything else commented out.
Take voltage measurements to see if you can find anything strange.