Is FNP = FEXPTIME if and only if NP = EXPTIME? It is very well known that if the classes $\sf FP$ and $\sf FNP$ are equal, then also the classes $\sf P$ and $\sf NP$ are equal (see e.g. FNP on Wikipedia). 

Is it also true that if $\sf FNP=FEXPTIME$ then also $\sf NP=EXPTIME$?  (See the exponential time conjecture.)  I did find a paper constructing real functions that are in $\sf FEXPTIME$.
Yes, vacuously, since it is trivial that $\operatorname{FNP} \neq \operatorname{FEXPTIME}$, because it takes an exponentially long amount of time to give an exponentially long output.