OLS bias in demand estimation: the bias always underestimate the demand's elasticity?
Usually, $\hat{\beta_1^{IV}} = \beta_1 + \frac{cov(z,u)}{cov(z,x)}$.  The denominator will go to zero. 

That is true unless there is some correlation between the instrument and the error term, and the nominator is the strength of the relationship between the instrument and the endogenous variable. The smaller the denominator gets, the greater the bias $\left[\frac{cov(z,u)}{cov(z,x)}\right]$.

In addition,  weak instrument will have no precision, so that the variance will have a big upward bias. 
\begin{eqnarray} var(\hat{\beta_1}) &amp;\to_p&amp; \frac{\sigma^2}{n \sigma^2_x} \nonumber \\
\hat{\beta_1^{IV}} &amp;=&amp; \frac{\sum (z_i - \bar{z})y_i}{\sum(z_i - \bar{z})x_i} = \beta_1 + \frac{\sum (z_i - \bar{z})u_i}{\sum(z_i - \bar{z})x_i} \nonumber \\
var(\hat{\beta_1^{IV}} &amp;=&amp; var \left( \frac{\sum (z_i - \bar{z})u_i}{\sum(z_i - \bar{z})x_i} \right) \nonumber\\
var(u | z) &amp;=&amp; \sigma^2 \nonumber\\
var(\hat{\beta_1^{IV}}) &amp;=&amp; \frac{\sigma^2 \frac{1}{n} \sum (z_i - \bar{z})}{n[ \frac{1}{n} \sum (z_i - \bar{z})(x_i - \bar{x})]^2} \nonumber
\end{eqnarray}

As $n \to \inf$
\begin{eqnarray} var(\hat{\beta_1^{IV}}) &amp;\to_p&amp; \frac{\sigma^2 \sigma_z^2 }{\sigma^2_{zx}} \nonumber\\
var(\hat{\beta_1^{IV}}) &amp;\to_p&amp;  \sigma^2 \frac{1}{n \sigma^2_x} \frac{1}{\rho^2_{xz}}\nonumber \\
\rho^2_{xz} &amp;=&amp; \frac{[\sigma^2 _{xz}]^2}{\sigma_x^2 \sigma_z^2} \textit{for} \rho \in [0,1]\nonumber
\end{eqnarray}

That is why if your instrument is weak, then you  may be better off running an OLS regression.