Is an WLS estimator unbiased, when wrong weights are used? It is clear that the WLS estimators are consistent if the "wrong" weights used aren't correlated with the explanatory variables. However, I don't know whether this also holds true for unbiasedness.
 A: It is unbiased, let's see:  Let the linear model be $Y=X\beta +e$, in matrix form, with $E e=0$ and the variance-covariance matrix of the errors $e$ be $\Omega$.  We use for weights the matrix $W$. Then the weighted linear least squares estimator is
$$
   \hat{\beta} = (X'WX))^{-1}  X'WX Y
$$
and we can calculate its expectation as
$$
    E \hat{\beta} = (X'WX)^{-1} X\cdot W E Y = (X'WX)^{-1} X\cdot W X\beta =\beta
$$
and you can observe that the variance-covariance matrix $\Omega$ do not play any role in the computations! 
Of course, if the weight matrix $W$ is estimated from the data in some way, then the above analysis is inadequate! This website  Engineering Statistics Handbook
discusses this problem, and gives references.  Scroll down to "Disadvantages of weighted least squares".  Their advice is:   

It is important to remain aware of this potential problem, and to only
  use weighted least squares when the weights can be estimated precisely
  relative to one another [Carroll and Ruppert (1988), Ryan (1997)].

