Let $f$ and $\beta$ be a scalar field and a $1$-form, respectively, on a 3-manifold $M$. The following is an equation for a $1$-form $\xi$:
$$\mathrm d \xi + \mathrm d f\wedge \xi = \mathrm d \beta \,,$$ or, setting $g=e^f$, $$\mathrm d (g \xi ) = g \mathrm d \beta \,.$$
Does it admit a solution for $\xi$?
It is probably very easy to see, especially if I knew EDS, but I don't... Thanks!
No EDS needed here. This is just basics. Assuming the $3$-manifold is simply connected (or else we can only answer locally), necessary and sufficient for $g\,d\beta$ to be exact is that it be closed. And it is closed iff $dg\wedge d\beta=0$, which holds iff $df\wedge d\beta = 0$. Without knowing anything more about your original $f$ and $\beta$, I can't say anything more.