First off, a pointer to literature: Hilbe, Negative Binomial Regression is the standard textbook on negbin regression. Very much recommended.
Now, the standard way of setting up a negative binomial regression (there are less common others) is via a log link, i.e., the mean $\mu$ is parameterized via a design matrix $X$ and coefficients $\beta$ as
$$ \log\mu = X\beta, \text{ or } \mu=\exp(X\beta). $$
I'll not distinguish between the true parameters $\beta$ and the estimates $\hat{\beta}$ in the following.
We can now distinguish different kinds of elasticity, depending on whether we are looking at a categorical/dummy predictor, or a numerical one.
Change in a dummy predictor
Assume we have $\mu_1=\exp(x_1\beta)$ and $\mu_2=\exp(x_2\beta)$, two fitted means, where the design matrix row vectors $x_1$ and $x_2$ differ in that one dummy flips from $0$ (in $x_1$) to $1$ (in $x_2$). That is, $x_2-x_1$ is a vector of $0$s and a single $1$ in the place at which the flipping predictor sits. Assume that this is the $j$-th entry. Then
$$\frac{\mu_2}{\mu_1} = \frac{\exp(x_2\beta)}{\exp(x_1\beta)}
= \exp\big((x_2-x_1)\beta\big) = \exp(\beta_j). $$
That is, flipping the $j$-th dummy from $0$ to $1$ will multiply the fitted mean by $\exp(\beta_j)$.
Change in a categorical predictor
Categorical predictors with $k$ factor levels are (again: commonly) represented internally as $k-1$ dummy predictors, for which the calculation above holds. Thus, switching from the reference category to the $j$-th (non-reference) category will change the fitted mean multiplicatively by $\exp(\beta_j)$. Changing from the $k$-th (non-reference) category to the $j$-th (non-reference) category will change the mean multiplicatively by $\exp(\beta_j-\beta_k)$.
Additive change in a numerical predictor
We again have to deal with design matrix row vectors $x_1-x_2$, which will again be a vector of $0$s except for the entry in which the changing predictor sits. So we can just assume that we have a single predictor that first takes a value $x$ (for a fitted mean of $\mu_1=\exp(x\beta)$) and then changes to $x+\Delta x$ (for a fitted mean of $\mu_2 = \exp\big((x+\Delta x)\beta\big)$). We obtain
$$ \frac{\mu_2}{\mu_1} = \frac{\exp\big((x+\Delta x)\beta\big)}{\exp(x\beta)}
= \exp(\Delta x\beta).$$
So changing a predictor by an additive $\Delta x$ will yield a multiplicative change of $\exp(\Delta x\beta)$ in the fitted mean, where $\beta$ is the coefficient corresponding to the changing predictor.
Multiplicative or percentage change in a numerical predictor
Assume a predictor's value changes multiplicatively, from $x$ to $cx$. As above, we get
$$ \frac{\mu_2}{\mu_1} = \frac{\exp(cx\beta)}{\exp(x\beta)}
= \exp\big((c-1) x\beta\big).$$
So in this case, the multiplicative change depends on the initial value $x$ of the predictor, in contrast to the effect of an additive change in the predictor, which per above does not depend on this initial value $x$.
Infinitesimal change in a numerical predictor
The economic definition of elasticity is the change in the effect as the predictor changes by an infinitesimal amount:
$$ \frac{\partial\mu/\mu}{\partial x/x} = \frac{\partial\mu}{\partial x}\frac{x}{\mu}. $$
For $\mu=\exp(X\beta)$, we have
$$ \frac{\partial\mu}{\partial x_j} = \frac{\partial}{\partial x_j}\exp(X\beta) =
\beta_j\exp(X\beta) = \beta_j\mu, $$
so
$$ \frac{\partial\mu}{\partial x_j}\frac{x_j}{\mu} =
\beta_j\mu\frac{x_j}{\mu} = \beta_jx_j. $$
Thus, the elasticity of $\mu$ with respect to the $j$-th predictor is $\beta_jx_j$ - so it again depends on the value $x_j$ of the $j$-th predictor at which we calculate the elasticity.
