Whether or not it makes sense to impute year of birth, and how to do it involves a number of considerations.
Firstly, imputation is probably only reasonable if the missingness pattern is Missing Completely at Random (MCAR) or Missing at Random (MAR). Some discussion of these missingness types are given in Section 25.1 of this paper. Ask yourself which type of missingness you are likely finding yourself confronted with. If you believe that there is a mechanism to the missingness you observe you may want to reconsider imputation.
Another question is do you consider birth year a categorical or a continuous variable.
If you believe it should be treated continuously, you can make use of a number of imputation methods. Multiple imputation may be one of the most appropriate. An bird's eye view is given here.
If you believe birth year should be treated categorically, you face the challenge of imputing a categorical variable. This is treated in this paper, which discusses the merits of a number of imputation procedures for categorical variables and provides some examples.
Multiple imputation for continuous and categorical variables can both be performed using the mi package in R.
So to summarize, you can impute birth year whether you want to treat it continuously or categorically. First though, think about whether there is a reason why those observations might be missing. Do you think they are MCAR or MAR, or can you imagine there is a systemic reason for their missingness? If so, is it Missingness that depends on unobserved predictors? If so, can you model the missingness somehow and prevent this from biasing you imputation? Is it Missingness that depends on the missing value itself? In either of the last two scenarios, you may want to think carefully about how to proceed with your analysis, and what conclusions you can reasonably draw from it.