Update: Things have gone from "Looks like a weird software bug" to "Damn, this could well be a serious hardware issue". As some users had been suggesting, the problem does indeed link to use of 2G. However, it turns out that the issue can be replicated by the use of 2G even on another, proximate phone. As you can see in the video demonstration by kongzs7 below, the volume rocker keys' sensors are set off even when the phone is only at the bootloader.
This absolves Android 4.0 from blame, and suggests either an issue with the internal hardware, or possibly the firmware of the device, both significantly more problematic to sort out.