MATLAB: Compute pole-zero diagram of FIR using only impulse response
Since your response is finite, your system corresponds to a FIR filter, where the coefficients are given by the values of the impulse response. Hence, you can perform Z-Transform of the FIR and find its zeros. Finally, since you would just need to calculate the zeros of the nominator polynomial. See this MATLAB code, following your impulse response example:

h = [1 0 -2 0 0 3];
b = h; % the FIR coefficients of a filter

roots(b)

zplane(b,1);


Output:

ans =

  -1.6379          
   1.2438 + 0.4498i
   1.2438 - 0.4498i
  -0.4248 + 0.9309i
  -0.4248 - 0.9309i




So, the zeros are directly given by the roots. For the poles, you have (N-1) poles at the origin.

Also, have a look at e.g. this answer.