Regressions with long-tail variables (GDP, etc) It seems common to apply standard linear regression to variables with long-tail distributions, like GDP, by first taking the log. What is the justification for doing that? Is it effectively assuming a lognormal distribution?
Also, if I have taken the log of an independent variable in a regression, how does that change the interpretation of $R^2$, p, etc?
 A: You don't need to assume a lognormal distribution; there's no requirement that an independent variable in linear regression itself has a normal distribution. The hope is that, with log transformation of the independent variable, the other requirements for interpreting linear regression results will better be met, such as having normally distributed residual errors independent of fitted values.
If the regression against the log-transformed independent variable meets those requirements, there are no problems with interpreting p-values, etc. Regression coefficients will now mean the change in the dependent variable per log change in the independent variable. So if you use log10, the regression coefficient will be "change per 10-fold change in GDP" for your example; for log2, "change per doubling of GDP."
A: 
long-tail distributions, like GDP,

Why do you think GDP has a long tailed distribution? This is not a common knowledge, as far as I know. 

by first taking the log. What is the justification for doing that? Is it effectively assuming a lognormal distribution?

Yes, we assume lognornal distribution, sometimes. If your log-transformed variable is normal, then the variable would be lognormal, but you don't have to assume normal distribution. 
It's still a debate whether GDP is a unit-root process or time-stationary though. For instance, some may think that $\ln GDP_i=A+Bi+\varepsilon$, i.e. time-stationary.
Log-transform is used most commonly for two main reasons. 
One is when you see that the variance of the series increases at higher levels. In this case often log or Box-Cox transformations are applied.
The second reason is when you think that your series have a constant growth rate, such as geometric brownian motion: $GDP_i=GDP_{i-1}e^r$, where $r$ is the growth rate. If you take the log, then this turns into a nice linear equation: $\ln GDP_i-\ln GDP_{i-1}=r+\varepsilon_i$. 

Also, if I have taken the log of an independent variable in a regression, how does that change the interpretation of R2, p, etc?

$R^2$ would be on fitting the log, i.e. your dependent variable would be $\ln\frac{GDP_i}{GDP_{i-1}}$, and all your OLS diagnostics would be on this variable.
Take a look at this paper: The Role of the Log Transformation in Forecasting Economic Variables, Helmut Luetkepohl, Fang Xu
