I mean, is there an antonym for "first-class function"?
First, the property "having first-class functions" is a property of a programming language, not of particular functions in it. Either your language allows for functions to be passed around and created at will, or it doesn't.
Functions that accept or return other functions as arguments are called higher-order functions. (The terminology comes from logic.) Functions that don't are called first-order functions. Perhaps this latter notion is what you wanted.
"First-class" here is roughly the informal use of the phrase meaning "having all rights and privileges", give or take. The alternative would be "second-class", as in the phrase "second-class citizen", but that seems to be less frequently used in this context.
It is, as Ohad says, a feature of a language which entities (functions or otherwise) it treats as "first-class", and it's also not necessarily a clear-cut distinction. Typically a first-class entity would be something tangible that can be created and used as a value at run time, without significant obstacles compared to being defined at compile time. In the case of functions, this usually means allowing higher-order and anonymous functions. A function that is not higher-order is first-order. Functions that are not anonymous don't really have a general term that I know of. But first-order functions with names can still be first-class if the language allows it.
In formal set theory a "mapping" which isn't a function (in the formal sense) is usually called a definable mapping. The reason this might be a meaningful term to import is that these are, in some sense, the things which behave like functions in the meta theory/language while not being, themselves, objects in the theory studied (the object language).
Alternatively borrowing terms from category theory one might want to call them external functions (since they are not "elements" of the internal Hom-objects, i.e. the exponential objects). (Kept this for the comments to make sense).
Borrowing the distinction from category between external and internal Hom-sets (that is to say collections of arrows of a category and exponential objects of a category) one might want to call "non-first class functions" external functions.
I doubt any of these are in common use, but they might at least be suggestive :)