How misleading is it to regard $\frac{dy}{dx}$ as a fraction?

You can think of $x$ and $y$ as smooth functions on a one-dimensional manifold of states of some system that you are thinking about, then $dx$ and $dy$ are differential forms. In any open region where $dx$ does not vanish we can say that $dy/dx$ is the unique smooth function such that $(dy/dx)dx=dy$; in other words, $dy/dx$ is $dy$ divided by $dx$. Of course you don't want to tell the students that, but it does clear up the logical question as asked.

[Added later:] this approach also gives a clear picture of what goes wrong with partial derivatives: if your state space has dimension $n>1$, then $dy$ and $dx$ lie in a vector space of dimension $n$, and you cannot divide them to get a number. I think it's a bit fussy to worry too much about notation for derivatives in one variable, but traditional notation for partial derivatives is horrendous, especially in any context where you might want to hold different variables constant in different places, such as Maxwell's relations in thermodynamics ( http://en.wikipedia.org/wiki/Maxwell_relations )

I am fine with using the notion of cancellation of fractions to help students remember the chain rule, but it is dangerous to be too cavalier with this idea. For example, suppose $F(x,y)=0$ defines $y$ implicitly as a function of $x$. Then $$\frac{dy}{dx}=-\frac{\frac{\partial F}{\partial x}}{\frac{\partial F}{\partial y}}.$$ Naive cancellation gives the wrong sign!

Treating dy/dx as a fraction is the gateway drug to treating ${\partial y}/{\partial x}$ as a fraction. This plus a little more notational confusion leads students to conclude that if $U(x,y)$ is a function of two variables, then along a level curve of $U$ we have

$$dy/dx = {\partial U/\partial x\over\partial U/\partial y}$$ by "cancelling the ${\partial U}$'s''.