Can fixed effects be "post-treament"? I have located a natural experiment in a time series cross-sectional dataset, but I am unsure of whether or not to include unit-level fixed effects in my models. I have produced a toy example that explains the situation: 
Say I have an experiment in which I observe some outcome for a group of units over time. Now, some are randomly assigned to receive a treatment at a specific time $t^*$. For two units, the data might look like this:
unit time outcome treatment

1    1       2         0
1    2       2         0
1    3       2         0
1    4       2         0
1    5       2         0
2    1       2         0
2    2       2         0
2    3       6         1      <- t^*
2    4       6         0
2    5       2         0

Say I then attempt to estimate the effect of the treatment by regressing $outcome$ on $treatment$, like so:
$outcome_{it} = \beta_0 + \beta_1treatment_{it} + \epsilon_{it}$
With this sample, $\beta_1 = 3.6$. But now, having variation over time for each unit, I might try to include unit-level fixed effects. The demeaned data looks like this:
unit time outcome treatment outcome.demean treatment.demean

1    1       2         0         0               0
1    2       2         0         0               0
1    3       2         0         0               0
1    4       2         0         0               0
1    5       2         0         0               0 
2    1       2         0       -1.6             -0.2
2    2       2         0       -1.6             -0.2
2    3       6         1        2.4              0.8
2    4       6         0        2.4             -0.2 
2    5       2         0       -1.6             -0.2 

Estimating the fixed effects model...
$outcome.demean_{it} = \pi_0 + \pi_1treatment.demean_{it} + \mu_{it}$ 
...yields $\pi_1=3$. A smaller effect. This is clearly  due to the fact that the effect of treatment seems to last two periods, the second of which is not marked by the treatment dummy, but is still included in the mean that is being subtracted from the outcome values of the treated unit. Thus I would argue that the fixed effects are "post-treatment" in this situation, because some of the values used in their calculation are consequences of the treatment without being incorporated into the treatment period. I would therefore opt not to include fixed effects here.
Am I correct in reasoning like this?
Note: Of course, the correct solution in this example would probably be to explicitly account for the lasting effect of the treatment by a lag or something equivalent, but in the actual setting where I encountered this problem, that is not feasible. 
Note 2:* Also, given the as-if-randomness of assignment, fixed effects should not be needed to obtain an unbiased estimate of the ATE. However, some people may be less convinced by the natural experiment than I am and might therefore call for fixed effects to be included.  
 A: Conditioning on post-treatment covariates produces a subgroup analysis in the same way that conditioning on baseline (pre-treatment) covariates produces a subgroup analysis.  There is nothing wrong with investigating subgroups; however, it is not necessarily the case that the causal treatment effect within a subgroup equals the treatment effect for the entire population.
If you have post-baseline events that interfere with the causal interpretation of the treatment for the entire target population I suggest you look at the ICH E9 addendum on estimands.  This is for clinical drug development but is useful in every setting.  Here is a related thread.
A: Your reasoning is not correct. First, if your experiment is randomized, treatment cannot be a (causal) function of any observables or fixed effects. So the fixed effects could indeed only be functions of the treatment, but I don't think that makes sense substantively. In any case, you would not need to adjust for the fixed effects, but you could benefit statistically from random effects.
Secondly, in your concrete example, the difference between the regression coefficients rests on the fact that they are also misspecified; by construction of the data, the treatment effect materializes over two periods.
