What is the effect of having a transition in an epsilon-NFA upon "$\epsilon$,0"?
I suggest you review the definition of epsilon-NFA, especially the definition of when an epsilon-NFA accepts a word.

To answer your question, no, the $0$ is not ignored. Consider an epsilon-NFA with two states, the initial state $q_0$ and the accepting state $q_1$, connected by two edges labeled $\epsilon$ and $0$. This epsilon-NFA accepts the words $\epsilon$ and $0$. If you remove the edge labeled $0$, the epsilon-NFA will accept only $\epsilon$.