You need not be concerned about upgrade unless you are DBA. It will be DBA's responsibility to properly configure 10g but as a developer you may have to do some regression testing on load/performance. If you know bit about the DB then you can look at the 8i & 10g configuration for DB mode, SGA size, max open cursor size, max connection and stuff like that. If you are java developer then noticeable difference is the 8i JDBC driver "oracle.jdbc.driver.OracleDriver" has been deprecated, in 10g drivers so you have to use "oracle.jdbc.OracleDriver"
Upgrading has the potential to completely break your code. That being said, there's really no way to know what will break and what won't without actual testing. What you normally do in a situation like this is perform the upgrade on a test system then perform a full set of system tests.
Its not the JDBC that is 'as likely' to fail but any stored procedurs, triggers, views, etc that your code depends on may stop functioning all together on an upgrade. I've seen upgrades from Oracle 9 to 10 that crashed JDBC code due to some dependence on a problematic function call in a stored procedure. For the most part, you just test it out on a copy of your database, figure out the issues, and if everything goes well, upgrade the real database. Last I checked Oracle lets you download developer versions of 10 for free so getting the software is easy enough.