High-resolution images show that Biden was adjusting the crease on his shirt and was wearing a rosary on his wrist. The claim is false.
In a high-quality video(34:37 minute), the mark on his shirt appears to be a crease, not a wire, which later disappears after a few seconds when Biden moves it. And Washington Reporter Matt Viser, responding to Trump's claim that Biden would not agree for ear inspection, tweeted that Biden was not wearing an earpiece during the debate.
In 2017, on Megyn Kelly Today show, Joe Biden had revealed that he wears his late son Beau's rosary beads around his wrist.
Biden was not wearing a wire in his debate with Trump, and therefore this claim is false.