EDITED text:
I was trying to get an answer to a slightly different problem, but I posted it here in greater generality, and made a slight mistake in it too. The question actually is, given that $f$ is continuously differentiable, and that $lim_{|t|\to \infty}f'(t)\exp(-t^2) = 0$ is $\int _{-\infty}^{+\infty}f'(t)\exp(-t^2)dt$ finite?
ORIGINAL text: Given $lim_{|t|\to \infty}f(t) = 0$ and that $f$ is continuously differentiable can we say $\int _{-\infty}^{+\infty}f(t)dt$ is finite?
Try the following function:
If $|t|\leq\frac\pi2$$$f(t) = \cos(t) + 2^{\frac{-1}4} $$
Otherwise $$f(t) = \frac1{|t| + 2^{\frac14} - \frac\pi2}$$
Why is this a counter example? Well, its easy to show this function is continuously differentiable at all points besides $t=\pm\frac\pi2$, and at these points, you can show through some algebra that this function is continuous and differentiable.
But, because of the nature of $\frac1t$, this function will have an infinite integral.
Now your question asks for $f'(t)e^{-t^2}$. We can set the function I found equal to this expression to find the actual $f$ that violates your condition.
Edit
My original example was $f'(t)\exp(-t^2) = \frac1{|t|+1}$, which happens to also be sufficient to disprove OP's question.