It is alright to have zero's when doing poisson regression. If there are more zeros in the data than predicted by a poisson distribution, than you can investigate this by fitting a zero-inflated poisson model. Alternatively, you can think about modeling the zero and >0 counts separately, using hurdle models.

I agree with Brian Gerber. In the latter case he mentions (the alternative of using hurdle models), this could be performed for instance by applying first logistic regression modeling to predict whether the response variable occurs or not, and an additional model (logistic or not) to predict the amount of counts (>0) once the response variable occurs.

Is it possible to omit the region that have zero counts in spatial analysis?

In my experience, region with zero counts are very interesting in the understanding of the problem at hand and you may want to consider other solutions (see good options by Sergio and Brian earlier) before exclusing these regions.  If you decide to exclude, be sure to do a good description of the characteristics of the regions excluded as compared to others, as this will be useful in discussing your results.  Another consideration is to ask yourself whether this is a zero because there is not yet a case, or because it is not possible to have a case (no exposure / infection / transmission possible).