There are some index issues with your expressions as mentioned by others. The correct way to raise/lower an index is by
$$ \partial^\mu = \eta^{\mu\nu}\partial_\nu .$$
Notice how the free index $\mu$ appears in the up position exactly once on each side of the equal sign, and the dummy index $\nu$ is repeated in the up and down position on the same side.
The dummy index $\nu$ is summed over, so the expression should be read as:
$$ \partial^\mu = \sum_{\nu=0}^3 \eta^{\mu\nu}\partial_\nu. $$
The expression tells you how to get each of the 4 components of $\partial^\mu$.
When we apply that to the relevant part of your second expression we get:
$$ \left(\partial_\mu \psi^*\right) \, \left( \partial^\mu \psi \right)
= \left( \eta_{\mu\nu} \partial^\nu\psi^*\right) \, \left( \eta^{\mu\lambda} \partial_\lambda \psi \right).
$$
To get the indices correct we make sure that each term in parentheses maintains a single $\mu$ index in the correct position, and that in a given multiplicative statement no index is used more than once in a given up/down position. These means we need a different dummy index for each summation, otherwise we might not know which terms get summed with which.
The final statement is then:
\begin{align}
\partial_\mu \psi \partial^\mu \psi^*
- \partial_\mu \psi^* \partial^\mu \psi
&= \partial_\mu \psi \partial^\mu \psi^*
- \eta_{\mu\nu} \partial^\nu\psi^* \eta^{\mu\lambda} \partial_\lambda \psi \\
&= \partial_\mu \psi \partial^\mu \psi^*
- \left( \eta_{\mu\nu} \eta^{\mu\lambda} \right) \partial^\nu\psi^* \partial_\lambda \psi \\
&= \partial_\mu \psi \partial^\mu \psi^* - \partial_\nu \psi \partial^\nu \psi^* \\
&= \partial_\mu \psi \partial^\mu \psi^* - \partial_\mu \psi \partial^\mu \psi^* \\
&= 0
\end{align}
We used the fact that
$$\eta_{\mu\nu} \eta^{\mu\lambda} = {\delta_\nu}^\lambda,$$
and that dummy indices can be renamed arbitrarily ($\nu\rightarrow\mu$).
Finally as a commenter stated, if you fully contract a metric with itself you get:
$$\eta_{\mu\nu} \eta^{\mu\nu} = {\delta_\nu}^\nu = 4$$
in a 4-D space.