All energy is conserved.
So if I push on an object and it doesn't move, where does that energy go?
That can only happen when you have static friction and in such a situation, the energy is converted to heat through friction and is distributed among different parts of the environment and the object. The whole energy is conserved but not the object's energy.
But when you say a force doesn't do work, it can mean something else too. You may have magnetic forces([itex] q\vec v \times \vec B [/itex]). It does no work because its always perpendicular to velocity. The particle's energy is conserved here.
Your body is not a very efficient machine. In this case, all of the energy you expend is used to tighten your muscle fibers, which stretch under the applied force. This means that all of the energy goes into stretching your muscles, eventually turning into heat, and none of it goes into moving the object.
Simply pushing on a stationary object doesn't give it any energy. It might feel like hard work but that's only for the reason others have given....namely that humans are not ideal machines. We are quite inefficient and consume energy even when not doing useful work. Much of that energy escapes as heat.
If you use a spring to push on the same stationary object you will find no energy leaves the spring. It all stays stored in the spring.
Static friction is not dissipating energy as heat. Sliding friction is.
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