In general, the push-forward of a line bundle on the flag variety $G/B$ will not be the direct sum of line bundles. (This holds only on toric varieties.)
However, Haboush [1] has proved that this is true for some special values of $\lambda$. More specifically, he proved that for $\lambda = \sigma + p\mu$ where $\sigma$ is half of the sum of the positive roots, one has an equivariant isomorphism $$ F_* \mathcal{O}(\lambda) \simeq V\otimes \mathcal{O}(\mu) $$ where $V$ is the Steinberg module, irreducible of highest weight $\sigma$. In particular, $F_*\mathcal{O}(\sigma)$ is a trivial vector bundle $V\otimes \mathcal{O}_{G/B}$ corresponding to the Steinberg representation.
[1] W. J. Haboush, A short proof of the Kempf vanishing theorem, Inventiones mathematicae volume 56, pages 109–112 (1980)