How does the naive Bayes classifier handle missing data in training?
In general, you have a choice when handling missing values hen training a naive Bayes classifier. You can either choose to either


Omit records with any missing values,
Omit only the missing attributes.


I'll use the example linked to above to demonstrate these two approaches.  Suppose we add one more training record to that example. 

Outlook  Temperature  Humidity   Windy   Play
-------  -----------  --------   -----   ----
rainy    cool        normal    TRUE    no
rainy    mild        high      TRUE    no
sunny    hot         high      FALSE   no
sunny    hot         high      TRUE    no
sunny    mild        high      FALSE   no
overcast cool        normal    TRUE    yes
overcast hot         high      FALSE   yes
overcast hot         normal    FALSE   yes
overcast mild        high      TRUE    yes
rainy    cool        normal    FALSE   yes
rainy    mild        high      FALSE   yes
rainy    mild        normal    FALSE   yes
sunny    cool        normal    FALSE   yes
sunny    mild        normal    TRUE    yes
NA       hot         normal    FALSE   yes



If we decide to omit the last record due to the missing outlook value, we would have the exact same trained model as discussed in the link. 
We could also choose to use all of the information available from this record.  We could choose to simply omit the attribute outlook from this record. This would yield the following updated table. 



           Outlook            Temperature           Humidity   
====================   =================   =================  
          Yes    No            Yes   No            Yes    No 
Sunny       2     3     Hot     3     2    High      3     4
Overcast    4     0     Mild    4     2    Normal    7     1 
Rainy       3     2     Cool    3     1
          -----------         ---------            ---------- 
Sunny     2/9   3/5     Hot   3/10   2/5    High    3/10   4/5 
Overcast  4/9   0/5     Mild  4/10   2/5    Normal  7/10   1/5
Rainy     3/9   2/5     Cool  3/10   1/5


            Windy        Play
=================    ========
      Yes     No     Yes   No
False 7      2       10     5
True  3      3
      ----------   ----------
False  7/10    2/5   10/15  5/15
True   3/10    3/5


Notice there are 15 observations for each attribute except Outlook, which has only 14. This is since that value was unavailable for the last record. All further development would continue as discussed in the linked article.  

For example in the R package e1071 naiveBayes implementation has the option na.action which can be set to na.omit or na.pass.