In general, the answer is no. Moreover, the answer is no even if 
\begin{equation}
	\phi(t)=t\ln(1+t). \tag{1}
\end{equation} 

Indeed, suppose that $P(Z_i=0)=1-2p$ and $P(Z_i=b)=p=P(Z_i=-b)$ for all $i$, where 
\begin{equation*}
	p:=\frac1{2\phi(b)},
\end{equation*}
$\phi$ is as given by (1), 
and $b$ is a large enough positive real number so that $p\in(0,1/2)$. 

Then for all $i$ we have $EZ_i=0$ and $E\phi(|Z_i|)=1$, so that $\|Z_i\|_\phi\le1$. On the other hand, for all real $c>0$ and all natural $n\ge2$
\begin{equation*}
\begin{aligned}
	&E\phi\Big(\Big|\frac1n\sum_{i=1}^n Z_i\Big|/c\Big) \\ 
	&\ge\sum_{i=1}^n \phi\Big(\frac b{cn}\Big)P(|Z_i|=b,\ Z_j=0\ \forall j\ne i) \\ 
	&=n \phi\Big(\frac b{cn}\Big)2p(1-2p)^{n-1} \\ 
	&=\frac{2pb}c\,\ln\Big(1+\frac b{cn}\Big)(1-2p)^{n-1}\to\frac1{2c}>1 
\end{aligned}
\end{equation*}
as $n\to\infty$, if $b=n^2$ and $c\in(0,1/2)$. So, for all large enough $n$ we have $E\phi\big(\big|\frac1n\sum_{i=1}^n Z_i\big|/c\big)>1$ and hence $\|\frac1n\sum_{i=1}^n Z_i\|_\phi\ge c$ and hence 
\begin{equation*}
	\Big\|\frac1n\sum_{i=1}^n Z_i\Big\|_\phi\not\to0 
\end{equation*}
as $n\to\infty$. 
 
---

More generally, the answer will remain no if $\phi(t)=t \ell(t)$, where $\ell$ is any function such that $\ell(t)$ is [slowly varying][1] as $t\to\infty$. Yet more generally, the answer will remain no if $\phi(t)=t L(t)$, where $L$ is any function such that $\sup\limits_{K\in(0,\infty)}\limsup\limits_{t\to\infty}\dfrac{L(Kt)}{L(t)}<\infty$. 

[1]: https://en.wikipedia.org/wiki/Slowly_varying_function