If X is a random variable with a normal distribution, then Y = exp(X) has a log-normal distribution.
Likewise if X is a random variable with a binomial distribution, then Y = exp(X) has a log-binomial distribution.
My question
In case of the log-normal distribution mean and variance are well known, for the log-binomial distribution I can't find any references.
We can use an entirely analogous technique to the one typically used to calculate the moments of a lognormal.
In particular, note that if $\newcommand{\E}{\mathbb E}X \sim \mathrm{Bin}(n,p)$ and $Y = e^X$, then $Y^k = e^{k X}$. But, $\E e^{kX} = m_X(k)$ where $m_X(t)$ is the moment-generating function of $X$ evaluated at $t$.
Hence, $$ \E e^{k X} = m_X(k) = (1 - p + p e^k)^n \>, $$ and so $$ \E Y = m_X(1) = (1 + (e-1)p)^n $$ and $$ \mathrm{Var}(Y) = m_X(2) - (m_X(1))^2 = (1+(e^2-1)p)^n - (1+(e-1)p)^{2n} \>. $$
Other moments of $Y$ can be computed in a similar fashion.
See: Parameter Estimation and Goodness-of-Fit in Log Binomial Regression, L. Blizzard and D. W. Hosmer, Biometrical Journal Volume 48, Issue 1, pages 5–22, February 2006
