Is it correct to say that p-values for Pearson's r decrease with increasing sample size because bias decreases with increased sample size? I realise there are other reasons why p-values will decrease with increased sample size, but I am wondering if this reason is valid at all.
 A: I think people may be answering or commenting on a question you didn't intend.
@ttnphns is correct (as usual) that the bias does decrease with increasing sample size. But this decrease is pretty small. $\sqrt{\frac{n-1}{n-2}}$ is small. If n goes from 10 to 100, this goes from $\sqrt{.9}$ to $\sqrt{.99}$ or from 0.949 to 0.995. If n is larger, the decrease is even smaller.
It is also true that the p value need not go down, as @Deathkill14 points out. 
However, in general the reason the p value goes down with increasing sample size (given the same correlation) is not because of decreasing bias but because of increasing precision. A simple random sample of a population gives an estimate that is nearly unbiased, and that "nearly" because unimportant for reasonable n. But it gets more precise much more quickly. An approximation to the standard error of the correlation is given by
$SE[r] = \frac{(1-\rho^2)^2}{\sqrt{n-1}}$
so, for $\rho^2$ = .5 when n goes from 10 to 100 (as above) this goes from 0.08 to 0.025 and this decrease continues for increasing n.
