I've posted all 787,444 solutions for $N=7$ at this link.
To compute these solutions, I've used bounds similar to those proposed in the answer by David desJardins, but only for terms $x_1, \dots, x_{N-2}$. The remaining two terms satisfy the equation: $$\frac{a}{c}+\frac{1}{x_{N-1}}+\frac{1}{x_N} = \frac{b}{c} \big(1-\frac1{x_{N-1}}\big)\big(1-\frac1{x_N}\big),$$ where $\frac{a}{c} := \sum_{i=1}^{N-1} \frac1{x_i}$$\frac{a}{c} := \sum_{i=1}^{N-2} \frac1{x_i}$, $\frac{b}{c} := \prod_{i=1}^{N-1} (1-\frac1{x_i})$$\frac{b}{c} := \prod_{i=1}^{N-2} (1-\frac1{x_i})$, and $\gcd(a,b,c)=1$, which is equivalent to $$\big((a-b) x_{N-1} + b+c\big)\cdot \big((a-b) x_N + b+c\big) = b(a-b) + (b+c)^2$$ from where the suitable values of $x_{N-1}$ and $x_N$ can be determined efficiently by factoring of the right hand side.
I've also added solution counts to the OEIS as sequence A369469. A similar sequence A369470 enumerates solutions with possibly equal terms.