We consider fractional stochastic heat equations with space-time Lévy white noise of the form $$\frac{\partial X}{\partial t}(t,x)={\cal L}_αX(t,x)+σ(X(t,x))\dotΛ(t,x).$$ Here, the principal part ${\cal L}_α=-(-Δ)^{α/2}$ is the $d$-dimensional fractional Laplacian with $α\in (0,2)$, the noise term $\dotΛ(t,x)$ denotes the space-time Lévy white noise, and the function $σ: \R\mapsto \R$ is Lipschitz continuous. Under suitable assumptions, we obtain bounds for the Lyapunov exponents and the growth indices of exponential type on $p$th moments of the mild solutions, which are connected with the weakly intermittency properties and the characterizations of the high peaks propagate away from the origin. Unlike the case of the Gaussian noise, the proofs heavily depend on the heavy tail property of heat kernel estimates for the fractional Laplacian. The results complement these in \cite{CD15-1,CK19} for fractional stochastic heat equations driven by space-time white noise and stochastic heat equations with Lévy noise, respectively.