Why can't HS- participate in E1 reactions? In my textbook (Organic Chemistry by Klein), it says that certain molecules can only act as nucleophiles and not as bases; thus they can only undergo substitution reactions, not elimination reactions. This group of molecules includes: $\ce{Cl- , Br- , I- , HS- , H2S , RS- , RSH}$. I know that while E2 reactions require strong bases, E1 reactions do not. Even a base like $\ce{H2O}$, which is pretty weak, can undergo E1 reactions. Then why can't something like $\ce{HS-}$ undergo a E1 reaction? For example, if we have a tertiary alkyl halide and $\ce{HS-}$, I would expect a mix of $S_N1$ and E1 products. However, my book claims there are only $S_N1$ products. Why is this the case?
As has been pointed out in the comments, the use of the word only is a little misleading. What it should really say is that $E1$ is possible but $S_N1$ is so strongly favoured that to all intents and purposes the only product is the $S_N1$ product.

This can be explained in terms of the relative energies and sizes of the orbitals involved. $\ce{HS-}$ has a large, diffuse, high energy HOMO which can have much better overlap with the empty $2p$ orbital of the carbon than the small, low energy $\ce{C-H}~\sigma^*$. Therefore, the $S_N1$ reaction is strongly thermodynamically favoured and dominates.

If you compare this to something like $\ce{H2O}$, the HOMO is much more compact and lower in energy and so it has much better overlap with the $\ce{C-H}~\sigma^*$ and so $E1$ is comparatively much more favourable.