Why is absolute time considered an axiom of Newtonian mechanics? What statements are based on this axiom? I guess absolute time is associated to classical mechanics because people like Newton believed in that concept, but are there actually any statements whose derivation is based on this assumption?
I've looked into the Noether's theorem and some proofs regarding the Lagrange equations and it didn't seem like this assumption was necessary.
 A: Absolute time means there is unique procedure to determine which events are simultaneous and this is present everywhere in Newtonian mechanics.
Take for example third Newton law. It states, that if object A acts on object B with some force, so does the object B on object A with force equal in magnitude and opposite in direction. But if the force is changing as system evolves, how do we know when exactly is the reaction force to be applied? The answer is, at the same time. I.e. there is an assumption that "at the same time" has absolute meaning, otherwise two different observers would assign reaction force at different events.

I've looked into the Noether's theorem and some proofs regarding the Lagrange equations and it didn't seem like this assumption was necessary.

How so? In newtonian mechanics, one demands Lagrangian to be invariant under Galilean transformations. These transformations do not transform time coordinate, so absoluteness of time is present right there in the symmetries of Lagrangian. I.e. if you start with list of all possible Lagrangians of Newtonian mechanics and find out their symmetries, you will notice that there is a class of coordinate transformations that keep all of them invariant and that these transformation can be used to define special kind of time. So it is there.
A: The concept of absolute time was perhaps more of a postulate, rather than an axiom.
If Newton's physics were based on any axioms, it would have to be the principle of relativity, which is perhaps the most profound concept, fundamental to his laws of motion. This principle was first stated by Galileo$^1$. It states that the laws of physics should be the same  in all inertial reference frames.
To this principle, Newton added his laws of motion, universal gravitation, and an assertion of an absolute time. He stated
"Absolute, true and mathematical time, of itself, and from its own nature flows equably without regard to anything external, and by another name is called duration: relative, apparent and common time, is some sensible and external (whether accurate or unequable) measure of duration by the means of motion, which is commonly used instead of true time.
So it is perhaps more accurate to not think of this absolute time as being based on an axiom, but rather a postulate, based on his (Newton's) understandings of space and time and other scientists understandings during that era.
$^1$ Newton's laws are invariant under Galilean transformations, described by the equations (for motion along the x-axis) $$x'=x-vt$$ $$y'=y$$ $$z'=z$$ $$\boxed{ t'=t }$$  Note that it becomes apparent that both Galileo and Newton (and others in that era) thought of time as absolute, encompassed in this last equation.
A: Newton and others did not think necessary to state that the time was absolute (at least not as a postalate$^1$), because they considered this self-evident. Indeed, if they doubted this, they would have come with something similar to (special) relativity much earlier.
I think the assumption about absolute nature of time is most obvious when deriving Galilean transformations as opposed to Lorentz transformations (I mean the pedestrian derivations where one adds velocities). You get the former one, assuming that the time is absolute, and the latter, assuming that the speed of light is constant.
Example: addition of velocities
As an example of a statement based on the absolute time we can take the addition of velocities. The position of an object in reference frame B is given by
$$
x_B(t) = x_A(t) + X(t),
$$
where $X(t)$ is the position of the origin of reference frame A. Assuming that the time is the same in the two reference frames, we can differentiate this equation, obtaining:
$$
v_B = v_A + V
$$

1 But see the Newton's quote in @josephh answer.
