For convenience, let $X$ be our space.
Specifically, can anyone name a few desirable properties or theorems that would fail if $X$ weren't required to be open? More generally, is there a part of topology that would completely fall apart?
It seems to me that we mainly want closure under arbitrary unions and finite intersections, which appears to be the more natural part of the definition (unlike "forcing" $\varnothing$ and $X$ to be open, which feels rather contrived). Of course, we get the empty set for free if we take an arbitrary union of nothing ($\bigcup \varnothing = \varnothing$), so that part really doesn't need to be in the definition.
Let's define a new word: A tolology on $X$ is a subset of $\mathcal P(X)$ that is closed under arbitrary unions and finite intersections. By the to(p/l)ological closure of a set I'm referring to the smallest to(p/l)ology containing it.
Let $\mathscr T$ be a topology on $X$ and consider $\mathscr T' = \mathscr T \setminus \{X\}$. If $\bigcup \mathscr T' = X$, then closure under arbitrary unions forces us to throw $X$ back in anyway, so the topological closure coincides with the tolological closure, nothing interesting here. Otherwise (this is what bothers me), we have $\bigcup \mathscr T' \subsetneq X$, then $\mathscr T'$ is still closed under arbitrary unions and finite intersections, so it's a tolology but not a topology. But throwing in $X$ adds nothing to the richness of the to(p/l)ology at all.
In fact, let's say $\bigcup \mathscr T' \subsetneq X$. Then the topological closure of $\mathscr T'$ is $\mathscr T$, but we still have a pretty boring space. For, if $\left|X \setminus \bigcup \mathscr T' \right| = 1$, then our space is not $T_1$, and if $\left|X \setminus \bigcup \mathscr T' \right| \geq 2$, then it's not even $T_0$. (Actually, any $T_1$ space must satisfy $\bigcup \mathscr T' = X$ by definition, and that probably covers just about every theorem in topology.)
Since un-requiring $X$ to be open doesn't give us any fewer theorems than we already have, and those spaces whose topology and tolology are different are as uninteresting as it gets, why can't we replace the definition of topology with that of tolology for simplicity's sake?
The only argument I can think of against this is that the first Kuratowski closure axiom says $\overline \varnothing = \varnothing$, so $\varnothing$ is closed, which means $X$ is open. But why do we need that first axiom?