Short answer: no. I'll give some context with the details of the simplest examples.
In the context of conservation laws, "energy" refers to the Hamiltonian. In classical mechanics, a quantity without explicit time dependence is conserved iff its Poisson bracket with the Hamiltonian is 0. In quantum mechanics, quantities are promoted to operators on a Hilbert space. The mean of a quantity without explicit time dependence is then conserved iff that quantity’s operator commutes with the Hamiltonian. (Quantities that commute with the Hamiltonian are called “good quantum numbers”.) Classical conservation laws thereby naturally survive quantisation; in particular, energy is conserved in both theories.
In the many-worlds interpretation, each universe uses its own “copy” of the Hamiltonian that is unrelated to what may happen in other universes. From the perspective of a physicist in either daughter universe, that universe existed before the wavefunction collapse, and quantities that classical mechanics predicts will be conserved are conserved in the above senses. And the total energy content of the parent universe may well be 0 anyway, in which case duplicating that universe would conserve even a multiverse energy measure. (Ditto with all other conserved quantities. For example, to within experimental error the universe has zero electric charge,)
According to particle-wave duality, an entity that is in some experiments a “particle” of energy $E$ and momentum $\mathbf{p}$ is in other experiments a “wave” of circular frequency $\omega=\tfrac{E}{\hbar}$ and wavevector $\mathbf{k}=\tfrac{\mathbf{p}}{\hbar}$. In truth neither of these classical interpretations are the whole story. The plane wave $\exp \text{i}\left(\mathbf{k}\cdot\mathbf{x}-\omega t\right)$ with $\text{i}^2=-1$ motivates the operator promotions $E=\text{i}\hbar\partial_t,\,\mathbf{p}=-\text{i}\hbar\boldsymbol{\nabla}$. (The former is the quantum-mechanical Hamiltonian discussed above.) In Newtonian mechanics, $E\psi=\tfrac{p^2}{2m}\psi +V\psi$ obtains the Schrödinger equation; in special relativity, $E^2\psi-c^2p^2\psi=m_0^2c^4\psi$ obtains the Klein-Gordon equation. (Including a potential energy term in the latter takes a bit more work.) In either case, the classical relation between energy, momentum and mass and associated conservation laws carry over naturally to a wave equation, solved by the above plane wave in the simplest cases.