  Vanadium dioxide (VO$_{2}$) undergoes a first-order metal-insulator
transition (MIT) upon cooling near room temperature, concomitant with
structural change from rutile to monoclinic. Accurate characterization of
lattice vibrations is vital for elucidating the transition mechanism. To
investigate the lattice dynamics and thermal transport properties of VO$_{2}$
across the MIT, we present a phonon renormalization scheme based on
self-consistent phonon theory through iteratively refining vibrational free
energy. Using this technique, we compute temperature-dependent phonon
dispersion and lifetimes, and point out the importance of both magnetic and
vibrational entropy in driving the MIT. The predicted phonon dispersion and
lifetimes show quantitative agreement with experimental measurements. We
demonstrate that lattice thermal conductivity of rutile VO$_{2}$ is nearly
temperature independent as a result of strong intrinsic anharmonicity, while
that of monoclinic VO$_{2}$ varies according to $1/T$. Due to phonon softening
and enhanced scattering rates, the lattice thermal conductivity is deduced to
be substantially lower in the rutile phase, suggesting that Wiedemann-Franz law
might not be strongly violated in rutile VO$_{2}$.
