As far as I understand not all experts in birational geometry would agree that Siu settles in his preprint Abundance conjecture, and this conjecture is considered for the moment as open. When Kawamata mentions in his recent paper the work of Siu
On the abundance theorem in the case $\nu=0$
his says that in his work Siu using analytic methods reproved the abundance conjecture for a non-minimal algebraic variety whose numerical Kodaira dimension is equal to 0. I would guess Kawamata would not write his paper if it were accepted that Siu gives complete proof of abundance conjecture.
So for the moment the answer to you question it should be "It conisdered that for $X$ of dimesnion $4$ and higher it is unknown " (in dimensions higher than three)if $H^0(nK_X)=0$ for all $n$ impies that $X$ is unirulled. As for analysis been stronger than algebra, there are examples when this is not quite the case. A famous one is bend and break, that uses characteristics $p$ and is a corner stone of minimal model program http://www.math.ens.fr/~debarre/Grenoble.pdf The existence of minimal models is still unknown even for 3-dimensional Kahler manifolds that are not algebraic...