Example 1 Find all non-right triangles $ABC$ such that
$$
\begin{array}{l}
\tan A \cdot \tan B \cdot \tan C \\
\leqslant[\tan A]+[\tan B]+[\tan C],
\end{array}
$$

where $[x]$ denotes the greatest integer not exceeding the real number $x$. (2009, Nanjing University Independent Admission Examination)
Given that in a non-right triangle $ABC$, we have
$$
\tan A \cdot \tan B \cdot \tan C
$$
$$
= \tan A + \tan B + \tan C.
$$

Therefore,
$$
\tan A + \tan B + \tan C
$$
$$
\leq [\tan A] + [\tan B] + [\tan C].
$$

Thus, $\tan A, \tan B, \tan C$ are all integers, and at least two of them are positive integers. Without loss of generality, assume $\tan A, \tan B$ are positive integers. Then from
$$
(\tan A \cdot \tan B - 1) \tan C = \tan A + \tan B,
$$
we know that $\tan C$ is also a positive integer. Therefore, we only need to find the positive integer solutions to the equation
$$
a + b + c = abc.
$$

Without loss of generality, assume $a \geq b \geq c$.
$$
\begin{array}{l}
\text{If } c \geq 2, \text{ then} \\
abc \geq 4a > 3a \geq a + b + c,
\end{array}
$$
which contradicts equation (1);
$$
\begin{array}{l}
\text{If } c = 1, \text{ then} \\
a + b + 1 = ab \\
\Rightarrow (a-1)(b-1) = 2 \\
\Rightarrow a = 3, b = 2.
\end{array}
$$

In summary, the positive tangent values of the three angles are $1, 2, 3$ for the triangle to satisfy the condition.