A derated takeoff may in fact save fuel - the takeoff takes longer but full rated Takeoff is typically the worst TSFC the engine ever runs at. So with a derated takeoff you're getting more thrust per unit of fuel.
Derated climb is a different story - while I suppose there are some specific circumstances where a derated climb might save a bit of fuel, as a general rule stilton is correct - getting to cruise altitude more quickly saves fuel.
Several years ago, we had a (new) 737-300 operator in Asia who was complaining that their CFM first run engines were running out of EGT margin after only 3,000-4,000 hours (typical was in the 12,000-15,000 hour range). Turns out they were flying short hops but running full rated balls to the wall every single takeoff
