Fixed Effects Estimation and Inconsistency Consider estimation of the following population regression function: 
$G{Y_{it}} = {\beta _0} + {\beta _1}IN{F_{it}} + {\beta _2}DE{M_i} + {\beta _3}POI{L_t} + {\varepsilon _{it}}$

Where:
$GY_{it}$ = percentage growth in real GDP per capita in country $i$ in year $t$

$INF_{it}$ = percentage rate of inflation in country $i$ in year $t$

$DEM_i$ = 1 if country $i$ is a democracy, 0 otherwise

$POIL_t$ = a price index of crude oil on the world market. 

If you were to explore estimation with fixed effects both for countries and for years of observation. Which coefficients in the above model could be estimated?
 Suppose that INF and POIL are correlated over time, and INF is correlated with DEM
across countries. Do the omitted variables in the fixed effects estimate result in
inconsistent estimates? Why? 

My guess is that the estimates are biased and consistent but I haven't been able to figure out why.
You can't identify the effect of oil price when Year FE are applied, since the world oil price is perfectly correlated with year Fixed Effects. 
You can't identify the democracy indicator if you country does not change its value in your observed period. 


For the other variables, it should be possible to obtain estimates. You should apply a fairly large sample to avoid collinearity issues (sth like 100 countries, 50 years). Further problems with your model are reverse causality, i.e. does growth cause inflation?