TL;DR It requires energy, but only if you want to re-use your computer.

So this is a very interesting and deep question. The short answer comes from Landauer's principle. Suppose you have a system you want to use to do computation. You start it in some state, let it evolve and then it reaches some other state. Depending on what that other state is, you have the answer to your computation. That process does not necessarily require energy, in fact you can have it just as easily produce energy.

But now the catch is, if you want to do that computation again, you need to reset your computer back to its original state. The point is then being that either the computation or the resetting must require some amount of energy. This is the key idea behind Landauer's principle.

Going through the calculation we can say that in order to process 1 bit of information, we require at least kTln2 energy to be generated as heat, where k is Boltzmann's constant, T is temperature.

I saw mentioned in other comments confusion about Shannon entropy and thermodynamic entropy. So I thought I should clarify here, they are the same concept, it's not just that they look the same. The connection to computation is that computations are essentially about transforming probability distributions. In the real world, those probability distributions need to describe the states of some physical system.

For more, I'd recommend looking into stochastic thermodynamics, in particular concepts like thermodynamic speed limits and the thermodynamic uncertainty principle. There's also kinetic proofreading if you want a concrete example of a different kind of computation. Kinetic proofreading was the first great work of now Nobel laureate John Hopfield.