How do I interpret a "difference-in-differences" model with continuous treatment? How do I interpret the ATE coefficient (i.e., the post-treatment indicator interacted with the continuous variable)? Does it make sense?
Should I break it down into subgroups and just run a fixed effects model instead (interact an indicator for each subgroup with post-treatment indicator)?
 A: Yes, it makes sense and in this case the coefficient for the interaction of the post-treatment indicator and the treatment variable gives you the effect on the outcome that results from an increase in the treatment intensity. An example of this is the paper by Acemoglu, Autor and Lyle (2004), where they estimate the effect of World War II on female labor supply in the US. In their model
$$y_{ist} = \delta_s + \gamma d_{1950} + X'_{ist}\beta + \varphi \left(d_{1950}\cdot m_s\right) + \epsilon_{ist}$$
$y$ are weeks worked by female $i$, in state $s$, in year $t$. They have two periods, 1940 and 1950 where $d_{1950}$ is a dummy for the latter year, $X$ is a vector of individual characteristics, $\delta_s$ are state dummies, and $m_s$ is the mobilization rate in each state. Their interaction estimates whether states with higher mobilization rates during WWII saw a stronger rise in females' weeks worked from 1940 to 1950. This is given by the coefficient $\varphi$.
This is also a difference in differences (DiD) setting with variable treatment intensity since mobilization rates $m_s$ are continuous and differ across states. They get a point estimate of 11.2 for $\varphi$, i.e. a 10 percentage points increase in the mobilization rate increased female labor supply by 1.1 weeks (note that their mobilization rate is between 0 and 100). States with higher treatment intensity therefore saw a bigger increase in female labor market participation as a result of the "treatment".
