Is approximate normality limited to the exponential family? A GLM looks like $$g(\mu) = X\beta,\ \ \mu = EY_i$$
where $Y_i$ is an exponential family.
It is commonly assumed for a decent sample size that $\beta$ is approximately normal with mean $\hat{\beta}$ and variance equal to the estimated covariance-matrix.
Question: Is this similarly valid for the same model, except where $Y_i$ is not an exponential family? Take a tweedie or a t-distribution as two common examples?
 A: This isn't restricted to the exponential family.
Under mild conditions, maximum likelihood estimates are asymptotically normal, with mean equal to the population parameter (which follows from consistency) and variance $I(\theta)^{-1}$, where $I(\theta)$ is the Fisher information (assuming the model is correctly specified).
See for example, the section on "Asymtptotic normality" in the Wikipedia article on maximum likelihood estimation. It outlines a proof there as well.
It's not always true (for an example of where it doesn't hold, it's not the case when the parameter is at the boundary of the parameter space).
So let's look at an example of a distribution that's not exponential family - the Laplace, say. The likelihood for its location parameter is maximized at the sample median, and the sample median is asymptotically normally distributed. The approach to normality appears to be quite rapid in this case:

So, to respond to the last part of the question, if we supposed that we had a regression-type model where the observations were independently and conditionally distributed from a $t_5$ distribution with location and scale parameters given by $X\beta$ and $\sigma$ then the ML estimate of $\beta$ for that model would indeed be asymptotically normal.
(Incidentally, if you fix the power parameter, $p$, in a Tweedie, it is exponential family and you can fit the mean using GLM. Similarly with a negative binomial. )

Going still further afield, you often get asymptotic normality even when not using maximum likelihood - many estimators may be shown to be asymptotically normal. 
