Question: For some reason I am suddenly confused over something which should be quit elementary.

In two-dimensional CFT's the two-point functions of quasi-primary fields are fixed by global $SL(2,\mathbb C)/\mathbb Z\_2$ invariance to have the form

$$\langle \phi\_i(z)\phi\_j(w)\rangle = \frac{d\_{ij}}{(z-w)^{2h\_i}}\delta\_{h\_i,h\_j}.$$
So a necessary requirement for a non-vanishing two-point function is $h\_i = h\_j$. Now consider the Ghost System which contains the two primary fields $b(z)$ and $c(z)$ with the OPE's

$$T(z)b(w)\sim \frac{\lambda}{(z-w)^2}b(w) + \frac 1{z-w}\partial b(w),$$
$$T(z)c(w)\sim \frac{1-\lambda}{(z-w)^2}c(w) + \frac 1{z-w}\partial c(w).$$
These primary fields clearly don't have the same conformal weight for generic $\lambda$, $h\_b\neq h\_c$. However their two-point function is

$$\langle c(z)b(w)\rangle = \frac 1{z-w}.$$

Why isn't this forced to be zero? Am I missing something very trivial, or are there any subtleties here?

Answer: 
1) Everything OP writes(v1) above his last equation is correct. The $bc$ OPE reads

$$ {\cal R}c(z)b(w) ~\sim~ \frac 1{z-w} ,$$

where ${\cal R}$ denotes radial ordering.

2) To calculate the two-point function

$$\langle c(z)b(w)\rangle $$ 

(which as OP writes must vanish if the conformal dimensions for $b$ and $c$ are different) is more subtle due to the presence of the ghost number anomaly, i.e. the vacuum should be prepared with certain modes of the $bc$ system, see e.g. Polchinski, *String Theory,* Vol. 1, Sections 2.5-2.7.
The answer seems to be that, technically at least, the two point function $\langle b(z) c(w) \rangle$ does vanish on the sphere. In the context of the standard $bc$ ghost system that shows up in bosonic string theory, the simplest nonzero correlation function on the sphere that involves both $b$ and $c$ is 
$$
\langle c(z\_1) c(z\_2) c(z\_3) c(z\_4) b(w) \rangle.
$$
This correlation function is a special case of (6.3.5) in Polchinski volume 1 and evaluates to 
$$
\frac{(z\_1-z\_2)(z\_1-z\_3)(z\_2-z\_3)(z\_1-z\_4)(z\_2-z\_4)(z\_3-z\_4)}{(z\_1-w)(z\_2-w)(z\_3-w)(z\_4-w)}.
$$ 

The Kronecker delta function $\delta\_{h\_i h\_j}$ that shows up in the result for $\langle \phi\_i(z) \phi\_j(w) \rangle$ described in OP's first formula can be argued for based on the transformations of the fields under inversion $z \to 1/z$. While $\langle b(z) c(w) \rangle$ cannot transform properly, as OP has noted, it would be reassuring to see that this five-point function does transform in the right way. Indeed, the conformal factor
$$
\frac{z\_1^2 z\_2^2 z\_3^2 z\_4^2}{w^4}
$$
takes precisely the right form to convert this correlation function into one involving only $1/z\_i$ and $1/w$.

(Note I am using the $SL(2, R)$ invariant vacuum state to compute these correlation functions. There is also the option of using so-called Q-vacua, some of which are obtained by acting with $c$ operators at the poles of the sphere. In this case, translation invariance is lost, and the correlation function $\langle \phi\_i(z) \phi\_j(w) \rangle$ can depend on a further dimensionless ratio $z/w$.)