In this paper, we investigate the scaling limit of heavy-tailed unstable cumulative INAR($\infty$) processes. These processes exhibit a power-law tail of the form $n^{-(1+\alpha)}$ for $\alpha \in (\frac{1}{2}, 1)$, and the $\ell^1$ norm of the kernel vector approaches $1$. We show that the discrete-time scaling limit also has a long-memory property and can be seen as an integrated fractional Cox-Ingersoll-Ross process.