One example that made the rounds on social media a few months back was a crew flying above 30,000 ft receiving "Terrain - Pull Up!" advisories. They ignored them, because there is no terrain on earth at 30k ft. If that had happened at a lower altitude, they would have likely had to do a terrain escape maneuver, which would definitely be felt by passengers, though it would likely not introduce a lot of additional safety risk.
The real risk comes when pilots begin to doubt that their safety systems are working correctly and thus ignore true warnings. Or when a bungled GPS leads pilots to fly into military airspace and get shot down.