Modular numbers I just learned about modular numbers on wikipedia, such as $17 \equiv 3\pmod{7}$.
So what is infinity $\pmod{n}$? It can't very well be all the numbers at once, so what happens?
 A: When we say $a \pmod n$, we need $a \in \mathbb{Z}$ and $n \in \mathbb{Z}\mathbin{\backslash}\{0\}$. So your question "$\infty\pmod n$" doesn't make sense in the first place. It is like asking "What is $\text{apple}\pmod n$?"
What you probably mean and want to know is "What is $\displaystyle \lim_\stackrel{x \in \mathbb{Z}}{x \to\infty} (x \bmod n)$?".
If $n \neq \pm 1$, then the answer is "It doesn't exist". If $n = \pm 1$, then the answer is $0$.
A: What do you mean by "infinity"?
Suppose you mean the projective numbers (specifically the projective integers, which is the same as the projective rational numbers), which adds a new point $\infty$ with algebraic properties such as $1/\infty = 0$,  $1/0 = \infty$, $\infty + 1 = \infty$, and the follwoing are undefined: $\infty + \infty$ and $\infty / \infty$.
If $n$ is prime, then we can also extend the integers modulo $n$ to the projective integers modulo $n$. Let's call the added element of that $\omega$.
In this case, we have $\infty \equiv \omega \pmod{n}$. We have other useful features too, such as
$$ \frac{1}{n} \equiv \omega \pmod{n} $$
so we can reduce any rational number modulo $n$ (if the denominator is relatively prime to $n$, we can define the reduction to be division modulo $n$ as usual).
If $n$ is not a prime, then things become trickier.

If you mean something else by "infinity"; e.g. the extended real numbers you see in calculus (i.e. $\pm \infty$), or the infinite cardinalities you see in set theory, or some other sort of thing, then asking to reduce it modulo $n$ is going to be nonsensical.
