Neither I nor he are going to date each other anymore.
we are not dating each other any more
might get used, grouping the "I am", "he is" as "we are". However, the condition can also be said as
Neither I nor he is...
Neither he nor I am...
where the verb is governed by the final noun/pronoun, though this may lead to certain formations sounding awkward. Since Sarah is talking about something in the future "will be dating" might be simplest to use.
By placing "I" first, Sarah is giving emphasis on herself. In school we are usually taught to put reference to ourselves at the end of a list for politeness. Obviously, Sarah is not happy.
Richard's response to Sarah should be
You're not (going to date each other anymore)?
Sarah: Neither I nor he will be dating each other anymore.
Richard: You won't?