The classical open mapping theorem in functional analysis certainly holds in the Banach space setting, and this is where I first encountered it. Slightly more advanced textbooks (e.g. Rudin's Functional Analysis) occasionally state and prove more general versions, for instance for $F$-spaces (i.e. metrizable by a complete translation-invariant metric, but not necessarily locally convex). Again, these qualify as classical results. It seems that the Banach property can be relaxed to a much weaker completeness condition, as long as the space is metrizable.
Even in the absence of metric, we still have a notion of completeness in general topological vector spaces (every Cauchy filter/net converges). However, the proof of the open mapping theorem for $F$-spaces relies on the Baire category theorem, so this really uses the complete metric in a non-trivial way. This leads me to the following question:
Question 1. Does the open mapping theorem generalize to arbitrary (not necessarily metrizable) complete topological vector spaces?
As the open mapping theorem comes in various forms (see e.g. Rudin Theorem 2.11), I may have to be a bit more specific. I am particularly interested in the following question:
Question 2. Are there complete topological vector spaces $X$ and $Y$ and a continuous linear surjection $T : X \to Y$ such that $T$ is not open?