There seems to be a connection between this question and the beautiful Karamata theory of regular variation.
A positive function $F$ is regularly varying both at $0$ and $\infty$ if and only if $F(t)=t^pL(t)=t^qK(t)$ where $p$ and $q$ are real and $L$ and $K$ are slowly varying at 0 and $\infty$ respectively, which implies in particular (due to the uniform convergence theorem for slowly varying functions) that $ \lim_{t\to0^+}\sup_{|a-1|\le \delta}\left|\frac{L(at)}{L(t)}-1\right| \to 0 \qquad \text{and} \quad \lim_{t\to\infty}\sup_{|a-1|\le \delta}\left|\frac{K(at)}{K(t)}-1\right| \to 0, $ for some $\delta>0$.
Now, it follows that whenever $f\sim g$ then $F\circ f \sim F\circ g$, provided that (i) $F$ is continuous, and (ii) $F$ is regularly varying both at 0 and $\infty$. The proof is a straightforward (but slightly tedious) $\frac\epsilon 3$ argument based on (a) the fact that $f(x)=g(x)(1+h(x)) \quad\text{ where }h(x)\to0,$ (b) uniform continuity of $F$ on any compact interval $[b,c]\subset(0,\infty)$, and (c) the limit relations above.
While conditions (i) and (ii) on $F$ are sufficient, (ii) is not necessary: The function $F(t)=t(2+\sin\ln t)$ is a counterexample. This function is not regularly varying since $\lim_{t\to0} F(at)/F(t)$ does not exist when $\ln a=\pi$. However, whenever $f\sim g$ it is true that $F\circ f\sim F\circ g$. This is due to the fact that $\sin(\ln g(x)+\ln(1+h(x))-\sin\ln g(x)\to0 \qquad \text{ whenever } h(x)\to0,$ by the uniform continuity of $\sin$.
Generalizing this example seems to yield a class that preserves $\sim$ consisting of all continuous $F$ such that $F(t)= t^p G(L(t))$ where $L$ is slowly varying at 0 and $G$ is bounded away from 0 and $\infty$ and is uniformly continuous, with a similar representation at $\infty$.
Naturally, the class of positive $F$ that preserve $\sim$ is closed under composition and products. I don't have any of the classic texts on regular variation handy, but it is conceivable that necessary and sufficient conditions are known.