I'm assuming you're talking about plane-polarized photons, where a photon that passes a 0º analyzer is horizontally polarized, a photon that passes a 90º analyzer is vertically polarized, and there's an orthogonal polarization basis at ±45º.
Here's the trouble:
Now whoever is at A measures A first at 0°, then (if it goes through), at 40°.
Like a river, you can never step in the same photon twice. Once you have analyzed the photon at 0º, you have irrevocably changed that photon to be either horizontally or vertically polarized. The probability that it passes a second polarizer is independent of whatever state it had when it was "born."
You can see this for yourself if you can find someone selling polarized sunglasses and borrow three pairs. If you take two linear polarizers and set them perpendicular to each other, the light transmission is zero. But if you take a third polarizer and put it in the middle, you can tune the total transmission of the system. With the middle polarizer at 45º you get 1/4 of the photons through: any information about their state in the horizontal-vertical basis gets destroyed when the pass through a diagonal polarizer.
Here's a quantitative analysis. Your polarization analyzers tell you whether a photon's linear polarization is parallel or perpendicular to a particular direction. If I have two polarizers whose axis are different by $\phi$, and they analyze your entangled photons, they'll see a correlation
$$
C = \frac{N_\parallel - N_\perp}{N_\parallel + N_\perp} = \cos 2\phi.
$$
This has the correct limiting behaviors: perfect correlation for $\phi=0$, perfect anticorrelation for $\phi=90º$, zero correlation for $\phi=45º$.
If the $N$ are fractions of the whole, $C$ is equivalent to
$$
N_\parallel = \cos^2 \phi, \quad\quad N_\perp = \sin^2 \phi.
$$
If A's primary analyzer is at 0º, and B's analyzer is at 20º, then A and B will compare notes after the experiment and find (neglecting experimental uncertainties) that their populations of photons fall into these four groups:
$$
\begin{array}
& & \parallel\text{ at B}
& \perp\text{ at B}
\\
\parallel\text{ at A} & 44.2\% & 5.8\% \\
\perp \text{ at A} & 5.8\% & 44.2\%
\end{array}
$$
Now add A's second analyzer.
The key insight is that A's first analyzer acts as a polarizer: once the photon is analyzed, its correlation to B is destroyed.
The transmission through A's second analyzer is given by Malus's Law,
$$
I = I_0 \cos^2 \theta,
$$
where $\theta$ is the angle between the two polarizers. For 40º, the transmission through the second polarizer is 58.7%. So the new results become
$$
\begin{array}
&
& & \parallel\text{ at B}
& \perp\text{ at B}
\\
\parallel\text{ at A and } & \parallel\text{ at A}' & 25.9\% & 3.4\% \\
\parallel\text{ at A and } & \perp\text{ at A}' & 18.3\% & 2.4\% \\
\perp \text{ at A} &
& 5.8\% & 44.2\%
\end{array}
$$
If B declines to make a measurement, or if B makes a measurement but his dog eats his notes, there is no effect on the transmission or polarization at A. Only when all the photon polarization measurements are compared can this pattern emerge.
Best Answer
Quantum teleportation requires a "classical channel" of information to be communicated between the two experimenters, so it doesn't violate the no-communication theorem because that theorem only rules out the possibility that two experimenters could communicate purely by their choice of measurements on parts of an entangled system. Referring to the schematic diagram of quantum teleportation below (from this page), the first experimenter performs a disruptive measurement on the system to be teleported (A) and also performs a measurement on one half (B) of a larger entangled system, then sends data on her measurements in some ordinary classical way (radio waves, an electrical cable, whatever--this is the "Send Data" arrow in the diagram) to the second experimenter, who then uses that data to perform just the right type of measurement on the other half of the entangled system (C) so that its state becomes identical to original state of A the moment before the disruptive measurement.