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I think that it can't since whenever we have data then we have mode and since we have mode it's frequency is at least 1. Therefore, variation ratio can't ever be 0.

What do you think?

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According to Wikipedia, the variation ratio is defined as

$v := 1 - \frac{f_m}{N}$

where $f_m$ is the frequency of the mode and N is the number of observations.

Imagine that there is 1 observation, and therefore 1 observation of the mode.

$$ \begin{eqnarray*} v&=&1 - 1/1\\ v&=&1 - 1\\ v&=&0 \end{eqnarray*} $$

So the variation ratio can be 0.

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