Note that $\mathrm e^{-\gamma C}\sum\limits_{k=0}^{C}\frac1{k!}(\gamma C)^k=\mathrm P(X_{\gamma C}\leqslant C)$, where $X_{\gamma C}$ denotes a Poisson random variable with parameter $\gamma C$. In particular, if $p\lt1$, the argument of the logarithm is between $1-p$ and $1$. Likewise, if $\gamma=1$, a central limit argument yields $\mathrm P(X_{C}\leqslant C)\to\frac12$. In both cases, the limit is zero.
From now on, assume that $p=1$ and that $\gamma\lt1$. One is interested in $ 1-\mathrm e^{-\gamma C}\sum\limits_{k=0}^{C}\frac1{k!}(\gamma C)^k=\mathrm P(X_{\gamma C}\gt C). $ Introduce some i.i.d. Poisson random variables $\xi$ and $\xi_k$ with parameter $1$ and, for every positive integer $n$, $\eta_n=\xi_1+\cdots+\xi_n$. Then, on the one hand $\eta_n$ is a Poisson random variable of parameter $n$ and on the other hand, for every $t\gt1$, the behaviour of $\mathrm P(\eta_n\gt tn)$ is described by a large deviations principle. More precisely, $ \mathrm P(\eta_n\gt tn)=\mathrm e^{-nI(t)+o(n)},\quad\text{where}\ I(t)=\max\limits_{x\geqslant0}\left(xt-\log\mathrm E(\mathrm e^{x\xi})\right). $ In the present case, $\log\mathrm E(\mathrm e^{x\xi})=\mathrm e^x-1$ hence $I(t)=t\log t-t+1$ for every $t\gt1$. Using this result for $n=\lfloor\gamma C\rfloor$ and $t=1/\gamma$, one gets $ \lim\limits_{C\to+\infty}-\frac1C\log\mathrm P(X_{\gamma C}\gt C)=\gamma I(1/\gamma)=\gamma-1-\log\gamma. $