Is the relativistic Dirac equation the most accurate way to calculate energy levels? I am curious about the current state of the art of calculating energy levels for atoms?  Is there a better way to do this now?
 A: 
Is the relativistic Dirac equation the most accurate way to calculate energy levels?

No. Quantum electrodynamics is famously more accurate.
Using the full Standard Model of particle physics would be even more accurate, but probably not noticeably enough to warrant the tremendous amount of additional labor.
A: It depends on which atom. QED corrections are necessary for the Hydrogen atom, or other two-body systems (deuterium, $^3$He$^+$, muonium or positronium), QED corrections are necessary to get energy levels accurate enough to account for the hyperfine splittings, Lamb shift, and the finite lifetime of the excited states. You may have a convenient overview of the current state of things in this review paper by S.G. Karshenboim. Notice that QED effects have to be introduced within carefully controlled approximations at the variance with Schrödinger's or Dirac's equation, exactly solvable for the Coulombic case, QED effects have to be introduced within carefully controlled approximations.
However, the world of atomic spectra (and the need for accurate calculations of energy levels) does not stop at the hydrogen-like problems. The remaining part of the periodic table is made of many-electron atoms. Polyelectronic atoms are a more complex challenge for theoretical calculations due to the electron-electron interaction requiring approximate, numerical methods even at the level of the non-relativistic treatment. Here, many methods are currently used, all based on numerical calculations, including QED effects in an approximate way. Just as an example, one can look at this quite recent paper.
