A recent ID question on an eavesdropping gone wrong evoked some memories in me of a similar scene in another movie, though with a different premise (and mood?).
It was about a police employee wiring some other officer for an undercover meeting. But he somehow misplaced the batteries of the microphone and during the undercover meeting that thing got hotter and hotter and ultimately either burned into the chest of the officer or created enough attention for the bad guys to discover it (ending equally badly for the undercover cop). This whole scene was recounted by the police employee responsible for the wiring in some personal failure flashback/story told to someone else. And at one point this guy had to finally overcome his low self-esteem due to this past event to yet again wire someone.
I have a strong feeling this had more of a comedic or satirical touch to it, or something told to be deeply serious while actually coming across as hilarious. That's why I first thought it fitted well into The Other Guys and its whole tone, but those guys have vastly different backstories. But when thinking more about it, it might as well have been a more serious story. Somehow I also tend to have the image of Chazz Palminteri (or somebody similar) on my mind as the guy being wired.
I'm 99% sure it's a US movie. But as to the age I'm a bit more unclear. Rather recent (say 10 years old) when comedic, or older (well, 90s or maybe even down to 80s) if more serious.