Here's Conor McGregor BOXING -- trading shots with a former IBO champion at a prestigious L.A. boxing gym ... and TMZ Sports has the footage.
It all went down at Box 'N Burn gym in Brentwood -- when the UFC superstar stepped in the ring with former IBO welterweight champ Chris van Heerden to see if he could hang.
Check out the video ... he takes some shots, he dishes some out ... but he still needs work if he plans to hang with Floyd Mayweather.