Unfortunately, this is not true.
Simple counterexample:
My original counterexample had some ugly numbers in it, but fortunately, there is a counterexample with nicer numbers. However, the explanation below might still prove informative.
Note that for $x>0$, $ f(x)=\frac{(x-1)(x-3)}{x^2+3}=1-\frac{4x}{x^2+3}\lt1\tag{1} $ Next, we compute $ f(2)=-\frac17\tag{2} $ Let $x_0=\frac{1}{256}$ and $x_k=2$ for $1\le k\le8$.
The product of the $x_k$ is $\frac{1}{256}\cdot2^8=1$, yet by $(1)$ and $(2)$, the sum of the $f(x_k)$ is less than $1-\frac87\lt0$.
Original couunterexample:
Let $x_0=e^{-3.85}$ and $x_k=e^{.55}$ for $1\le k\le 7$.
We get $f(x_0)=0.971631300121646$ and $f(x_k)=-0.154700260422285$ for $1\le k\le 7$.
Then, $ \prod_{k=0}^7x_k=1 $ yet $ \sum_{k=0}^7f(x_k)=-0.111270522834348 $
Explanation:
Let me explain how I came up with this example.
$\prod\limits_{k=0}^nx_k=1$ is equivalent to $\sum\limits_{k=0}^n\log(x_k)=0$. Therefore I considered $u_k=\log(x_k)$. Now we want $ \sum_{k=0}^nu_k=0 $ to mean that $ \sum_{k=0}^n\frac{(e^{u_i}-1)(e^{u_i}-3)}{e^{2u_i}+3}\ge0 $ I first looked at the graph of $\large\frac{(e^{u}-1)(e^{u}-3)}{e^{2u}+3}$. If the graph were convex, the result would be true.
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Unfortunately, the graph was not convex, but I did note that $f(u)$ dipped below $0$ with a minimum of less than $-\frac17$ near $u=.55$, and that it was less than $1$ everywhere. Thus, if I took $u=.55$ for $7$ points and $u=-3.85$ for the other, the sum of the $u_k$ would be $0$, yet the sum of the $f(e^{u_k})$ would be less than $0$.