Let $\{h_n(x)\}_{n\in\mathbb{N}}$ a family of infinitely differentiable functions, defined on the entire real line, vanishing outside $[a,b]$. Let $F(x)$ be any function defined on $\mathbb{R}$ with the property that $F$ and all its derivatives are $O(|x|^{-N})$ as $|x|\to\infty$, for every $N$.
By hypothesis, the following limit:
$$\lim_{n\to\infty}\int_{x=a}^b h_n(x)F(x)\mathrm{d}x$$
exists for any $F$ of the type specified above.
Is it true or false that $\lim_{n\to\infty}\int_{x=a}^b h_n(x)F(x+\tau)\mathrm{d}x$ converge uniformly w.r.t. $\tau\in[\tau_1,\tau_2]\subset\mathbb{R}$?
I tried with Cauchy criterion, but without success... If the answer is "no", with which hypothesis the answer could become "yes"?
As said in the comments of your post, the hypothesis on $F$ toward infinity is useless. We can choose $F$ to be any smooth function defined over $\mathbb{R}$. But since only the values of $F$ on a neighborhood of $[a,b]$ are important, we can assume that $F$ has a compact support if needed.
I can answer positively to your question if we add the hypothesis that the sequence $\|h_n\|_{L^1} = \int_a^b |h_n(x)| dx$ is bounded. In this case, by uniform continuity of $F$ (it can be assumed compactly supported thus Heine theorem holds), $\tau \mapsto (x \mapsto F(x + \tau))$ is continuous for the $L^\infty$ norm (just play with $\varepsilon$-$\delta$ and the definition of uniform continuity). If we set for all $n$ and $\tau$, $$ I_n(\tau) = \int_a^b h_n(x)F(x + \tau) dx $$ and $I(\tau) = \lim_{n \rightarrow +\infty} I_n(\tau)$ (which exists by hypothesis), we have for all $n,\tau,t$, $$ |I_n(t) - I_n(\tau)| \leqslant \left|\int_a^b h_n(x)(F(x + \tau) - F(x + t))\right| \leqslant \|h_n\|_{L^1}\|x \mapsto F(x + \tau) - F(x + t)\|_{L^\infty}. $$ By continuity of $\tau \mapsto (x \mapsto F(x + \tau))$ and boundedness of $(\|h_n\|_{L^1})$, we deduce that the $I_n$ are uniformly continuous, thus its limit $I$ is continuous, which is the wanted result.
In the general case, the question is harder. For example, if you take $h_n(x) = n^k\cos(2\pi(b - a)nx)$ on some $[a + \varepsilon,b - \varepsilon]$ that you extend into a smooth function supported in $[a,b]$, you can show that your hypothesis holds for any smooth $F$ and any $k \geqslant 0$ (look at the convergence speed of Fourrier coefficients) but if $k > 0$, the hypothesis I added is not verified.
Try to see if this particular case is a counter-example, it may help you to find a counter-example or a proof in the general case.