  Weak lensing (WL) promises to be a particularly sensitive probe of both the
growth of large scale structure (LSS) as well as the fundamental relation
between matter density perturbations and metric perturbations, thus providing a
powerful tool with which we may constrain modified theories of gravity (MG) on
cosmological scales. Future deep, wide-field WL surveys will provide an
unprecedented opportunity to constrain deviations from General Relativity (GR).
Employing a three-dimensional (3D) analysis based on the spherical
Fourier-Bessel (sFB) expansion, we investigate the extent to which MG theories
will be constrained by a typical 3D WL survey configuration including noise
from the intrinsic ellipticity distribution $\sigma_{\epsilon}$ of source
galaxies. Here we focus on two classes of screened theories of gravity: i)
$f(R)$ chameleon models and ii) environmentally dependent dilaton models. We
use one-loop perturbation theory combined with halo models in order to
accurately model the evolution of matter power-spectrum with redshift in these
theories. Using a Fisher information matrix based approach, we show that for an
all-sky spectroscopic survey, the parameter $f_{R_0}$ can be constrained in the
range $f_{R_0}< 5\times 10^{-6}(9\times 10^{-6})$ for $n=1(2)$ with a 3$\sigma$
confidence level. This can be achieved by using relatively low order angular
harmonics $\ell<100$. Including higher order harmonics $\ell>100$ can further
tighten the constraints, making them comparable to current solar-system
constraints. We also employ a Principal Component Analysis (PCA) in order to
study the parameter degeneracies in the MG parameters. Our results can
trivially be extended to other MG theories, such as the K-mouflage models. The
confusion from intrinsic ellipticity correlation and modification of the matter
power-spectrum at small scale due to feedback mechanisms is briefly discussed.
