This paper I'm reading claims that the answer to my question is yes, but I'm having trouble finding another source.
I know that a poisson distribution tends to a normal distribution as the mean becomes large, and that the product of independent poissons has a mean equal to the product of the independent poisson means, which could become large as N becomes large, but I don't think the product of poissons is itself poisson so I'm not sure the using the normal limit is appropriate.
Answering the question in title: no.
The product of positive variables tend to converge to log-normal distribution, since in log space we have central limit theorem. However, that is not the case for Poisson distribution, because it always has a finite probability of zero. So as the $N\to\infty$ the probability that one of the multiplicants is zero and this nullifies the whole product goes to $1$.
However, the paper states something different. It's not about convergence of Poisson distributions, it's about the convergence of posteriors, which is Gamma-distribution. And it's not about multiplication of random variables, but rather multiplication of distributions (otherwise you would have seen a lot of convolutions).