Flex vs Rated Thrust
 

Someone told me that utilizing a flex (derated) thrust will result in a higher fuel consumption compared to burnage with rated thrust settings for takeoff .

I always thought that it would take more energy to achieve a faster acceleration in subsonic conditions . Can anyone elaborate or point me in the right direction please.

If you spend longer to accelerate more gradually in thick air, then you potentially use more fuel to get to the same speed.

Acceleration is proportional to thrust, but energy expended is the time integral of thrust and velocity, so depends on the speed profile as drag builds on takeoff, and then needs to be tweaked for the relationship between thrust and fuel consumption.

Just along the runway, it seems reasonable that reaching the same take off speed more slowly will consume more fuel, as while the aircraft has the same energy when it leaves the ground, when derated there's a longer track of moving air stirred up behind you.

However, is the motivation for derating is about saving money by protecting the engine from unnecessarily high temperatures? In that case, a few kg of fuel is cheaper than an overhaul.

Considering the ROC is less, you spend more time getting to altitude. This MAY (depending on the engines' SFC curve) burn more fuel - again the tradeoff is longer time on wing for your engines.

Taken to the extreme, use idle power for takeoff and you will burn all of your fuel without ever getting airborne.

Derated takeoffs & climbs
 

According to Boeing:
On repeated full-thrust takeoffs and climbs, the parts that suffer the most would be the combustor, hot section seals and HPT airfoils. Below is an interesting presentation given by Rolls Royce on this subject:

Derated Climb Performance