Does the electrochemical series reverse in electrolysis?
No, only the first is. The oxidising or reducing property of a species is tied to the species, not to the process in which the species is used. Therefore regardless of whether you are running a galvanic cell or an electrolytic cell, $\ce{F2}$ will always be a strong oxidant and $\ce{Li}$ a strong reductant.

The only difference about the two types of cells is that in the galvanic cell, you are letting the spontaneous cell reaction happen, which produces energy. In the electrolytic cell, you are using energy to force the non-spontaneous cell reaction to happen. The spontaneous cell reaction refers to the direction of the reaction with $\Delta G &lt; 0$ (or equivalently, $E &gt; 0$).

By this, I mean: you can use a galvanic cell to carry out the reaction $\ce{2 Li + F2 -&gt; 2Li+ + 2F-}$. Since $\ce{Li}$ is a strong reductant and $\ce{F2}$ is a strong oxidant, this reaction will generate a lot of energy.

You can use an electrolytic cell to reverse that reaction (although you'd have to use molten and not aqueous $\ce{LiF}$). But since $\ce{Li+}$ is a weak oxidant and $\ce{F-}$ a weak reductant, you're going to need lots of energy to carry that out. Which makes perfect sense!