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The construction of the total ring of fractions shows that any commutative ring (with 1) can be put inside a ring whose non-units are zero divisors.
Examples of these rings are the products of fields.
Are there other examples? Do these rings have a standard name?
Following Lam's Lectures on modules and rings these would be called commutative classical rings because they are equal to their localizations at their set of regular elements. (See section 11 in the book.)
That is, all the regular elements are already units: therefore something which isn't a unit has to be nonregular, i.e. a zero divisor.
The broadest, easily described subclass, I think, is that of strongly $\pi$-regular rings, which are rings that satisfy the DCC on all chains of the form $xR\supseteq x^2R\supseteq x^3R\supseteq\cdots$ for every $x\in R$.
That includes all one-sided artinian rings and all von Neumann regular rings, covering all examples given in the comments and in the original post.
Suppose, though, that you fabricate strongly $\pi$-regular rings $R_i$ such that there exists $x_i\in R_i$ satisfying $x_i^iR\neq x_i^{i+1}R$. Then in $R=\prod_{i=1}^\infty R_i$, it is easy to see that nonunits are zero divisors, but $R$ is not strongly $\pi$-regular, because the element $x=(x_1, x_2,\ldots)$ spoils things. So at least we can say rings with your condition are a strictly bigger class than the strongly $\pi$-regular rings.
