A ResultSet object is automatically closed by the Statement object that generated it when that Statement object is closed, re-executed, or is used to retrieve the next result from a sequence of multiple results.
The init.ora parameter open_cursors defines the maximum of opened cursors a session can have at once. It has a default value of 50. If the application exceeds this number the error "ORA-01000: maximum open cursors exceeded" is raised.
Therefore it's mandatory to close the JDBC resources when they are not needed any longer, in particular java.sql.ResultSet and java.sql.Statement. If they are not closed, the application has a resource leak.
In case of reusing the Connection object, you must be aware of the fact that the opened oracle cursors are kept open and in use as long the connection exists and the transaction has not ended. When the application commits, the opened cursors are released.
Therefore as an application designer you need to know a rough estimation of the needed open cursors for your most complex transaction.
The difficulty lies in the inability of oracle's internal parameter views (v$open_cursor, v$sesstat, et. al.) to show the difference between opened cursors, which are reusable and opened cursors, which are still blocked (not reusable!) by an unclosed ResulSet or Statement. If you close all Statement and ResultSet objects in your finally block, your application is perfectly fine.
Adjusting the init.ora parameter works like this (our application needs 800 cursors at a maximum)