Will fluoride evaporate from tap water at the same rate as chlorine
Short answer: no, fluoride does not evaporate in the same way/rate as chlorine.

Longer answer: chlorine will be present as covalent, molecular $\ce{Cl2}$. This is, when not in solution, a gas. Chlorine molecules leaving the surface of the water will (on average) not come back (at least unless there is a very high concentration of $\ce{Cl2}$ in the room's air, in which case you have other problems).

Fluoride, on the other hand, is present as $\ce{F-}$ ions (or other, complex ions) along with corresponding positively charged ions (usually $\ce{Na+}$, I believe). Even if the $\ce{NaF}$ were to be removed from solution, $\ce{NaF}$ is a non-volatile salt, and so would not evaporate away.