I'm not sure if this is true but I have the intuition it should be and I can't find a counterexample. So far I've figured out $f$ must be nonnegative in all the domain (otherwise the integral goes to $-\infty$ because the function is monotone decreasing).
I tried to lower-bound the integral by an infinite sum of smaller integrals in an interval of length $n$ where the function could be bounded by $f(n)$, but the lenghth of the interval grows slower than the number for which I can bound the function. See, for example, how $\int_{0}^{3}f(x)\,\mathrm{d}x\geq\int_{0}^{1}f(x)\,\mathrm{d}x+\int_{1}^{3}f(x)\,\mathrm{d}x\geq f(1)+2f(3)$. On the other hand, $\int_{0}^{\infty}f(x)\,\mathrm{d}x=\lim\limits_{x \to \infty}\int_{0}^{x}f(y)\,\mathrm{d}y\geq\lim\limits_{x \to \infty}xf(x)$, but this only guarantees that the limit is finite, not $0$.
