Oscilloscope causes a problem with my envelope demodulator
It's hard to be certain without knowing the physical arrangement of the circuit, but one possibility is that connecting the scope ground creates a ground loop that interferes with the circuit operation:

One key characteristic of this circuit is that it has a lot of gain. The input-capacitor to feedback-capacitor ratio in the first stage gives a gain of (2.2 nF) / (27 pF) = 81 in the passband. In the second stage, the resistor ratios are setting the gain in the passband to (62K) / (2.2K) = 28, which leads to an overall voltage gain of about 2270.

Connecting the scope ground clip can create a closed electrical path that goes through the circuit ground, through the power supply to earth ground, and finally through the scope ground back to the circuit ground. Many sources of noise and interference can drive a circulating current on this ground loop: (a) If the ground path goes through the digital portion of the circuit, digital switching currents can inject noise current into the ground. (b) An external source of magnetic interference (a CRT, radio, or poorly shielded inductor or transformer) can induce current in the ground loop. (c) If the scope and circuit power supply plug into different receptacles, current already present in the building grounding conductors can drive current through the ground loop. (d) Even the scope can be a source of ground currents (although hopefully not very much for a good scope).

In any case, the flow of unwanted ground current can generate small voltages across the non-zero impedance of the ground conductors in the amplifier circuit. The large gain of this circuit can amplify even a few millivolts of noise or interference on the amplifier's ground to a level that causes the envelope detector to trip the MCU input.

One thing you could try is to connect the scope ground clip to a point farther from the amplifier circuit. Imagine the path that any current from the ground clip will take back to earth ground, and keep that ground current from flowing through the amplifier ground wires or ground pane. The scope waveforms may appear noisier, but an alternate grounding point may eliminate the unwanted operation.

At higher frequencies, a ground loop can exist even without a DC ground loop. A filter capacitor between grounds can close a ground loop -- even the inter-winding capacitance of a transformer can allow the flow of high-frequency ground currents. This means that the unsafe practice of "floating the scope" by using a cheater plug won't necessarily break a ground loop -- but it will break the safety ground.