Can I use fixed effects regression with non-time-variant treatment? I have data on a number of units, some of which have received treatment and some of which haven't. Binary outcome data was collected for each unit at multiple times. The model is therefore
$$\text{logit}(y_{it}) = \beta_0T_i + \beta_1U_i$$
where $T_i$ is treatment and $U_i$ is the unit indicator (i.e., unit fixed effect). I would like to measure the contribution of treatment on my outcome outside of any unit-specific effects. The outcome $y_{it}$ varies within units but the treatment does not.
Can I still use fixed effects regression?

Unfortunately, $T_i$ will be dropped.
According to your specification, $T_i$ is time-constant. Say you're observing individuals $i$ over time. In all periods $t$, any individual $i$ is either in the treatment group or they're not. Thus, there's no within-person time variation to exploit. The $i$-level fixed effect removes all time-invariant heterogeneity across $i$, which invariably includes a dummy variable denoting group membership.
Your equation is estimable so long as there's observations of each $i$ before and after some shock (i.e., treatment). In that case, you'd interact $T_i$ with a second term demarcating before versus after the event. But if there's no time $t$ where a shock hits some units and not others, then you cannot proceed with this approach.
In short, you'd have to drop the unit fixed effect to achieve something estimable, or at the very least exploit some timing impact where some treatment/stimulus varies across $i$ and $t$.
