  In this article, we study the Euler class of taut foliations on the Dehn
fillings of a $\mathbb{Q}$-homology solid torus. We give a necessary and
sufficient condition for the Euler class of a foliation transverse to the core
of the filling solid torus to vanish. We apply this condition to taut
foliations on Dehn fillings of hyperbolic fibered manifolds and obtain many new
left-orderable Dehn filling slopes on these manifolds. For instance, we show
that when $X$ is the exterior of a pretzel knot $P(-2,3,2r+1)$, $r\geq 3$,
$\pi_1(X(\alpha_n))$ is left-orderable for a sequence of positive slopes
$\alpha_n$ with $\alpha_0 =2g-2$ and $\alpha_n\to 2g-1$. Lastly, we prove that
given any $\mathbb{Q}$-homology solid torus, the set of slopes for which the
corresponding Dehn fillings admit a taut foliation transverse to the core with
zero Euler class is nowhere dense in $\mathbb{R}\cup \{\frac{1}{0}\}$.
