I seems that in general that information is not conserved as opposed to energy, so any direction relationship seems unlikely but perhaps it is expressed in the quantum or other non-classical realm of physics?
Actually, Scott Aaronson gave a clear (if heuristic) argument for this in his essay on whether "information is physical" is contentful. He has an argument for the Bekenstein bound, stating that the amount of information inside a sphere of radius $R$ with total energy $E$ is less than $kRE$ for some big constant $k$. Scotts argument (some shortening on my part):
Relativity ... demands that, in flat space, the laws of physics must have the same form for all inertial observers (i.e., all observers who move through space at constant speed).
Anything in the physical world that varies in space—say, a field that encodes different bits of information at different locations—also varies in time, from the perspective of an observer who moves through the field at a constant speed.
- Combining 1 and 2, we conclude that anything that can vary in space can also vary in time. Or to say it better, there’s only one kind of varying: varying in spacetime.
- More strongly, special relativity tells us that there’s a specific numerical conversion factor between units of space and units of time: namely the speed of light, c. ...
- Anything that varies across time carries energy. Why? Because this is essentially the definition of energy in quantum mechanics! Up to a constant multiple (namely, Planck’s constant), energy is the expected speed of rotation of the global phase of the wavefunction, when you apply your Hamiltonian. ... No energy means no looping around means nothing ever changes.
- Combining 3 and 5, any field that varies across space carries energy. More strongly, combining 4 and 5, if we know how quickly a field varies across space, we can lower-bound how much energy it has to contain.
- In general relativity, anything that carries energy couples to the gravitational field. This means that anything that carries energy necessarily has an observable effect: if nothing else, its effect on the warping of spacetime. ...
- Combining 6 and 8, any field that varies across space couples to the gravitational field.
- More strongly, combining 7 and 8, if we know how quickly a field varies across space, then we can lower-bound by how much it has to warp spacetime. This is so because of another famous (and distinctive) feature of gravity: namely, the fact that it’s universally attractive, so all the warping contributions add up.
- But in GR, spacetime can only be warped by so much before we create a black hole: this is the famous Schwarzschild bound.
- Combining 10 and 11, the information contained in a physical field can only vary so quickly across space, before it causes spacetime to collapse to a black hole.
Note that this shows that if you want to have information stored in a field - and in quantum field theory this is all that is available - then there is going to be a required amount of energy and space to keep it. The converse might not be true: you could perhaps have a lot of energy that contains no information.
As for conservation, energy is only locally conserved in general relativity, while in the general case it is not conserved. Information is conserved in quantum mechanics because the evolution of the Hamiltonian and whatever unitary operators you apply are all time-reversible. In practice information tends to leak and apparently randomise when quantum systems touch their environment, and since spacetime is at least approximately time-invariant energy looks pretty conserved.
Landauer's principle expressed as "there is an energy cost to erasing information", is strictly speaking not true. You can erase a bit without paying an energy cost by exchanging it from a fresh zeroed bit from some pre-made empty computer memory, or by randomizing a spin or other conserved quantity that is in a no entropy initial state. But since such resources are rare, we typically have to pay energy to pump their random information into the background heat-bath of the universe, so practically the principle implies that thermodynamics demands a bit erasure cost.
Are these "fundamental" relationships? I don't know. But information requires a physical basis to be stored in, and such a basis will have energy.