  In this paper, a necessary and sufficient condition is obtained for the scale
invariant boundary Harnack inequality (BHP in abbreviation) for a large class
of Hunt processes on metric measure spaces that are in weak duality with
another Hunt process. We next consider a discontinuous subordinate Brownian
motion with Gaussian component $X_t=W_{S_t}$ in ${\bf R}^d$ for which the
L\'evy density of the subordinator $S$ satisfies some mild comparability
condition. We show that the scale invariant BHP holds for the subordinate
Brownian motion $X$ in any Lipschitz domain satisfying the interior cone
condition with common angle $\theta\in (\cos^{-1}(1/\sqrt d), \pi)$, but fails
in any truncated circular cone with angle $\theta \leq \cos^{-1}(1/\sqrt d)$, a
Lipschitz domain whose Lipschitz constant is larger than or equal to
$1/\sqrt{d-1}.$
