I have a multinomial logistic regression model. One of the output categories is not observed in the data set that I'm using.
If you are talking about a missing value in the response, there are many available texts on imputation with specific tailoring to regressions. I have not read it, but Frank E. Harrell's text is tailored to logistic regressions and has a chapter on missing values.
If a potential outcome has never been observed, then you have no information about the effect of covariates on it. So any outcome-specific covariate effects are unidentifiable. If you assume constant covariate effects, then this outcome has no effect on their estimation, so you might as well omit it from the data.
More generally, you do have some information about the frequency of that outcome, but logistic regression (multinomial or otherwise) cannot handle this at all, because if you insist on having this outcome, the corresponding intercept has to be $-\infty$.
In business practice, this comes up a lot.
For example, in credit scoring say you included a set of marketing sources as a variable, odds are new partners are going come into the pipeline after you created the score.
In general, with infrequent categorical levels, say those with under $n$ examples, we recode them as "Else" and make that the reference category in the logistic regression
Pragmatically speaking the real question is how frequently do new categoricals appear and what proportion of the population are they.
If they are infrequent you can code them as "Else". This is not mathematically optimal but from a rubber hits the road standpoint it functions as a workaround.
If they are high volume however, you should need to create a new score to include them.
