Create an input so that poles show up as outputs?
You should always be able to achieve this after $N-1$ samples. For this I will denote the given transfer function as

$$
H(z) = \frac{A(z)}{\prod_{k=1}^N (z - p_k)}. \tag{1}
$$

Now by starting with an impulse and filter it with

$$
F(z) = \frac{\prod_{k\neq i} (z - p_k)}{z^{N-1}} = \sum_{k=0}^{N-1} \alpha_k\,z^{-k} \tag{2}
$$

and use the output of this filter as the input to the system. The combined transfer function looks like

$$
H(z)\,F(z) = \frac{A(z)}{(z - p_i)\,z^{N-1}}. \tag{3}
$$

Therefore this filtered signal should only excite the mode of the system associated with the pole $p_i$. And the $z^{N-1}$ term only adds a delay of $N-1$ samples, so it only takes a finite amount of time after which only $p_i^n$ is visible in the output.

However it can be noted that this does require you to have perfect knowledge of the poles you cancel.