From the completeness relation one can see that, $$|\psi \rangle = \int \frac{d^2 \alpha}\pi \langle \alpha | \psi \rangle |\alpha\rangle.$$ And if $|\psi\rangle = |\beta \rangle$ (which is another coherent state), then $$|\beta \rangle = \int \frac{d^2 \alpha}\pi \langle \alpha | \beta \rangle |\alpha\rangle = \int \frac{d^2 \alpha}\pi \exp(-\frac{|\alpha|^2}2-\frac{|\beta|^2}2+\alpha^{*}\beta) |\alpha\rangle.$$
From this above expression how can one come to the conclusion that the coherent states are not linearly independent and thus over-complete.