I assume here that your device is not in the feedback chain.
If you can't afford a FFT or filter-bank decomposition (and then detect over successive frames the FFT bins in which the amplitude gets almost exactly multiplied by the same complex number over successive frames), I would suggest looking at these few parameters:
- Fit a line to the log of the RMS envelope (computed on 50 or 100ms windows) over the past 2 seconds. Compute the slope, compute the coefficient of correlation. First should be positive, the second should be close to 1 for a feedback build-up, which is exponential.
- Compute the standard deviation of the zero-crossing intervals. This is a cheap harmonicity measure, and it should go close to zero as a pure tone settles in and overwhelms the signal.
- % of clipped samples. Seeing clipping is a bad omen - but maybe it's too late for you?
You should get by with a few decision rules based on these criteria.
Now, if your device is in the feedback loop (eg: audio DSP box, hearing aid), things are easier because you can "challenge" what you think is a feedback frequency by inducing a delay in the processing chain, or more subtly an all-pass filter (to delay only the suspicious frequency), and if you notice a change in the frequency of the suspect peak, you get a confirmation that it was induced by feedback rather than coming from the input signal.