If you choose a GAM with a Poisson distribution and log-link, it means that
- the response is modelled by a Poisson random variable $Y$,
- the mean $E[Y]$ is related to the nonlinear predictor $\sum f(x_i)$ via the link function, i. e.
$$log\left(E[Y]\right) = x_0 + \sum f(x_i) \Leftrightarrow E[Y] = exp\left(x_0 + \sum f(x_i) \right)$$
A Poisson random variable can be zero with positive probability, i. e. there is no reason to exclude these observations. In that sense, $y=0$ is not different from $y=1$ or any other value. There is no log transformation of the response variable.
Side note: If there are weights for each observation (if you don't specify them, they are equal to $1$ for all observations), these weights would be log-transformed into an offset
$$E[Y] = W \cdot exp\left(x_0 + \sum f(x_i) \right) = exp\left(log(W) + x_0 + \sum f(x_i) \right). $$
Here, observations with weight zero have to be excluded. Weights can be used when trying to model ratios. An example is modelling the claims frequency in insurance, where $Y$ would be the number of claims and $W$ the number of insurance contracts.
