It is not more efficient. If you're in a Boeing put '0' in the cost index and it will display the most efficient speed.
Minimum climb rate is 500' FPM(U.S. standards). Lower climb rate requires notification to ATC.
The most efficient place to increase speed, and decrease time, with the lowest additional fuel burn is on the descent. The least efficient is increasing cruise speed. If you're on a Boeing you can verify this by modifying the cost index and seeing what impact increasing cost index does to the CLB/CRZ/DES speeds. DES is increased first, then CLB speeds, and it takes amazingly high cost index numbers to increase the cruise mach on the FMC.
His statement that the time saving pays for the gas is wrong. You should be able to verify that by crossing checking the time saved vs. fuel used at the first waypoint after top of climb.
Another issue is why do you care what he does? It's not unsafe, you know it's not true, and you say you can't get through to him. If you can't get through to him, like he won't listen to facts that disagree with how he views the world, why bother worrying about him? Mark it down as a lesson for when you upgrade to Captain and the value of being technically correct when you pass on knowledge to your FO's.