What's wrong with this reasoning that $\frac{\infty}{\infty}=0$? 
$$\frac{n}{\infty} + \frac{n}{\infty} +\dots = \frac{\infty}{\infty}$$

You can always break up $\infty/\infty$ into the left hand side, where n is an arbitrary number. However, on the left hand side $\frac{n}{\infty}$ is always equal to $0$. Thus $\frac{\infty}{\infty}$ should always equal $0$.
 A: How do you know $0+0+0+0+...... = 0$?  If you think about it $0 + 0 + 0 +..... = 0\times\infty = 0\times 1/0 = 1$. (or any other number).  We clearly are dealing with a value on which standard arithmetic doesn't apply. (Hence "infinity is not a number".)
So the question becomes what does apply and how do we deal with this?  And that is not an easy/simple question.  It's not hard... but it's not simple.  Bottom line, finite rules of arithmetic do not always apply, and such instincts lead to common traps.
A: When you learned how to extend arithmetic from the natural numbers to the integers to the rational numbers to the real numbers and to the complex numbers, one of the main motivations was to preserve the laws of arithmetic.
The situation with the extended real numbers (i.e. the real numbers along with $\pm \infty$) is different — the goal is not to preserve the laws of arithmetic, the goal is to have continuity.
Consequently, arguments that involve naively applying the ordinary laws arithmetic to extended real numbers are quite unreliable.
This is compounded by the fact the argument involves infinite summation. Infinite sums can fail to satisfy many of the laws that finite sums obey, so that's a second source of unreliability.
A: The existing answers are very good; let me give yet another one. 
Consider the counter-argument that ${\infty\over\infty}=\infty$: We have $\infty=2^\infty$, so $${\infty\over\infty}={\infty\over 2^\infty}={\infty\over 2}\cdot {1\over 2}\cdot {1\over 2}\cdot {1\over 2}\cdot...$$ But ${\infty\over 2}$ is infinity, and stays infinity no matter how many times we divide it by $2$. So ${\infty\over \infty}=\infty$. 
This is exactly the same "shape" as the argument you give, but yields the opposite answer; so something must be wrong with this kind of argument.
The problem lies in the "number" $\infty\over\infty$. It hasn't been precisely defined. Now, some of the time in math it's clear what we mean when we write some complicated expression; however, this isn't one of those times. We need to sit down and precisely define what this thing is.
When we try to do so, we'll find something surprising: it doesn't follow the usual laws of arithmetic! For example, consider the following "proof" that $1=2$: $${1\cdot{\infty\over\infty}}={\infty\over\infty}={2\infty\over \infty}=2\cdot{\infty\over\infty},\mbox{ so }1=2.$$ Since $1\not=2$, at least one of the steps there has to be nonsense. The obvious candidate is the claim that we can cancel $\infty\over\infty$. This suggests that ${\infty\over\infty}=0$; however, then we have $\infty\cdot {1\over\infty}\not=1$, which ruins one of the basic properties of division! 
It gets worse: in both arguments, we need to perform infinitely many operations (either infinitely many sums, or infinitely many products), so we need to define how those work. At first glance it seems like limits can help us there, but again, we'll find that there's no good way to define the infinite operations we need which matches with our intuition.
This all points to the following fact:

In order to make sense of arithmetic involving infinity or infinite operations, we need to make some choices in how we precisely define the various operations on infinity; and no matter how we make these choices, some "obvious" properties of numbers will fail to hold.

Picture trying to fit a carpet into a room that's too small. You can maybe get lots of it to lie flat, but somewhere it's going to fold over, or stick up, or crumple. 
A: I think an issue with your reasoning as well is that to "split up infinity" you would have to have an infinite sum of finite terms. You can see with Riemann integration that an infinite sum of 0s is not necessarily 0. Of course none of this is rigorous but I think it should help you intuition a little.
A: Infinity is a fine number (and so is its negative counterpart), and is surprisingly well-behaved if you unpack it carefully.
Without going into too much detail about what that means, just note that Cantor showed that there are many types of infinities with different characteristics.
This means that we have to be clear about what sort of infinity we're talking about.
Most of the existing answers implicitly interpret infinity as some number implied to be the result of some unbounded number of arithmetic accumulations, which makes sense given the framing of your question. So what kind of infinity is reached in that fashion? $\omega$!
$\omega$ is a number that, informally, is always bigger than any real number you can produce using finite arithmetic steps. It's a member of the hyperreals, symbolized as $\mathbb{R}^*$. The nice thing about the hyperreals is, anything you can state in first-order logic (i.e., no tricks with arguments on sets) about the reals is still true. This is known as the transfer principle.
In the hyperreals are also infinitesimals $\epsilon$, which are the reciprocals of the infinitudes. They work just as you'd expect: a real number divided by an infinitude is an infinitesimal. So the hyperreal number line looks like this (Wikipedia):

With these tools, we note that each of the terms in your series are infinitesimal:
$$\frac{n}{\infty} = \epsilon$$
You have an infinitude of them, so the series can be rewritten as:
$$\epsilon + \epsilon + \dots = \omega\epsilon$$
This is an indeterminate form in the hyperreals (see chapter 1 of Keisler), just as you would expect. The reason for this is because $\frac{n}{\infty}$ is not zero, and in the first place, it wouldn't make sense for a infinitely divisible real number to be zero in that case.
So you can have infinity and infinitesimals as your equation depicted, but you still have to acknowledge indeterminates if you do so.
A: You cannot use an "infinite" distributive law with division by the symbol $\infty$. In other words the statement
$$\frac{\sum_{n=1}^\infty a_n}{\infty} = \sum_{n=1}^\infty \frac{a_n}{\infty}$$
is not valid if $\sum_n a_n$ is a divergent series. For the left-hand side cannot have any meaning when that sequence diverges, whereas the right-hand side is the zero series (provided that each $a_n$ is an ordinary finite number) which has the sum $0$ for any reasonable summation definition.
The law $\frac{a+b}{d}=\frac{a}{d}+\frac{b}{d}$ is true for real or complex numbers $a$, $b$ and $d$ provided $d\ne 0$, but it does not quite generalize to the form you use.
A: The problem is that before you can even consider a proof of a statement, it has to be clear what that statement means. I have no idea what $\infty/\infty=0$ could possibly mean.
The original answer ended here.
Since not everyone was happy with that answer, let me elaborate. Since $\infty$ is not a number, I just do not know what $\infty/\infty$ should mean. Now you could respond that you are using formulas like $\infty+\infty=\infty$ and have been told that that one is true. Well, it is a useful short-hand, but what people really mean when they write this is the following:
If $(a_n)$ and $(b_n)$ are sequences of real numbers such that $\lim_{n\to\infty} a_n=+\infty$ and $\lim_{n\to\infty} b_n=+\infty$, then $\lim_{n\to\infty} a_n+b_n=+\infty$.
Now this is a perfectly fine statement, and it happens to be also true. Another perfectly fine statement (but a wrong one) would be:
If $(a_n)$ and $(b_n)$ are sequences of real numbers such that $\lim_{n\to\infty} a_n=+\infty$ and $\lim_{n\to\infty} b_n=+\infty$, then $\lim_{n\to\infty} a_n/b_n=0$.
Now maybe one could understand $\infty/\infty=0$ in this way, but you would have to say so. And now you also see that it would be hard to see how the “proof” in this question relates to this. Maybe one could make the following weaker statement:
If $(b_n)$ be a sequence of real numbers such that $\lim_{n\to\infty} b_n=+\infty$, then $\lim_{n\to\infty} n/b_n=0$.
One could then say that the “proof” in your question is to be understood as follows: 
\begin{multline}\lim_{n\to\infty} \frac{n}{b_n}
=\lim_{n\to\infty}\underbrace{\left(\frac1{b_n}+\cdots+\frac1{b_n}\right)}_{\text{$n$ summands}}
=\\=\lim_{n\to\infty}\underbrace{\left(\lim_{n\to\infty} \frac1{b_n}+\cdots+\lim_{n\to\infty}\frac1{b_n}\right)}_{\text{$n$ summands}}
=\\=\lim_{n\to\infty}\underbrace{(0+\cdots+0)}_{\text{$n$ summands}}=\lim_{n\to\infty}0=0.
\end{multline}
Now here at least every expression has a defined meaning and we can ask: Where is this wrong? Well, it turns out that the second equality is wrong, and the reason why is again boring: There is just no reason that this equality should hold. It is not enough that it somehow looks nice, we would have to cite some theorem that we have proved earlier to justify this equality, and there is none. (Of course we can pinpoint the wrong equation by plugging in a counterexample. For example for $b_n$=n you can evaluate each expression and get $1=1=0=0=0=0$.)
Anyway, my point is that before we can ask why something is wrong it first has to make minimal sense. And the proof in your question is, as they say, “not even wrong”, because it is unclear what the statement is.
A: First of all $\infty$ is not a real number, so it's unclear what you mean by $n/\infty$ in the first place.
But, let's suppose that you extend the real numbers by introducing a new number "$\infty$" which has some expected properties, such as $n/\infty = 0$ for any real number $n$.  You would still need to define $\infty/\infty$ before you can refer to it, otherwise we don't even know what "$\infty/\infty$" means.  You could define it to be $0$ if you liked, but somebody else could equally well argue that $\infty/\infty$ should be defined to be equal to $10$ (for example) because
$$
\lim_{x \to \infty} \frac{10x}{x} = 10.
$$
Another person could argue that $\infty/\infty$ should be defined to be equal to $\infty$, because
$$
\lim_{x\to\infty} \frac{x^2}{x} = \infty.
$$
There is no value that we could assign to $\infty/\infty$ that stands out as being more valid than any other possible value.  So, we simply leave $\infty/\infty$ undefined.
A: If $\frac\infty\infty=0$, then we should have $\infty\cdot 0 = \infty$.
But OP's main equation $\frac{n}{\infty}+\frac{n}{\infty}+\frac{n}{\infty}+\cdots=\frac\infty\infty$ (which is supposed to equal $0$) suggests that $\infty\cdot 0 =0$.
How can $\infty\cdot 0$ equal both $0$ and $\infty$?
