If I have a logistic regression model with 3 predictors, $x_1$, $x_2$, $x_3$, and then I remove $x_3$ from my model (left with only $x_1$ and $x_2$), are those models nested? And therefore I can use the likelihood ratio test ($\chi^2$ with 1 df in this case) to test if $x_3$ is necessary to keep in the model. Or are these models not nested because they do not include the exact same variables?
Yes those models are nested. Quoting wikipedia's Statistical_model entry: Nested models are models that can be obtained by restricting a parameter in a more complex model to be zero. which is exactly what is happening when you "remove" $x_3$.
Additionally: Assuming you want to use Akaike information criterion (AIC - probably one of the most commonly used Information Criteria using likelihood ratios), you don't even need to worry about that. AIC works with non-nested models also.