  We study the pyrochlore series (Eu$_{1-x}$Bi$_x$)$_2$Ir$_2$O$_7$ for $ 0 \leq
x \leq 1$. We show that for small $x$, the lattice undergoes an anomalous
contraction but the all-in/all-out and metal-to-insulator transitions remain
robust, and the resistivity approaches a $1/T$ dependence at low-T, suggesting
proximity to the Weyl semimetallic phase, as previously predicted
theoretically. At the boundary between Eu$_2$Ir$_2$O$_7$ and Bi$_2$Ir$_2$O$_7$
a qualitatively different ground state emerges, which is characterized by its
unusual metallic behavior and absence of magnetic ordering at least down to
$0.02$ K.
