We can use the log sum property $\sum \log_b a_i = \log_b (\prod a_i)$ to simplify the sum: $\log_{10} (\tan 1^\circ \cdot \tan 2^\circ \dots \tan 89^\circ)$.

Recall the trigonometric identity $\tan(90^\circ - \theta) = \cot \theta$. This allows us to pair terms: $\tan 1^\circ \cdot \tan 89^\circ = \tan 1^\circ \cdot \cot 1^\circ = 1$. Similarly, $\tan 2^\circ \cdot \tan 88^\circ = 1$, and so on up to $\tan 44^\circ \cdot \tan 46^\circ = 1$. The middle term left unpaired is $\tan 45^\circ = 1$.

Therefore, the entire product inside the logarithm simplifies to $1$: $\log_{10}(1) = 0$.