Yes, the condition $EY_i=0$ cannot be dropped. Indeed, if e.g. the $Y_i$'s are iid with $\mu=EY_i\ne0$, then, by the law of large numbers, the left-hand side of the inequality in question will go to $1$ for each $\epsilon\in(0,\|\mu\|)$, whereas the right-hand side of the inequality will go to $0$.

The condition $EY_i=0$ cannot be dropped. 

Indeed, if e.g. the $Y_i$'s are iid with $\mu=EY_i\ne0$ and $n\to\infty$, then, by the law of large numbers, the left-hand side of the inequality in question will go to $1$ for each $\epsilon\in(0,\|\mu\|_V)$, whereas the right-hand side of the inequality will go to $0$.