Do OR conditions in Oracle SQL queries spoil index use / cause full table scan?
I had a very similar issue in Oracle 12c. In experimenting, the PRECOMPUTE_SUBQUERY optimizer hint 'forced' the subquery to execute first, thus making use of the defined indexes.  In your snippet, that would look like this:

...
where t.ObjectId = 12345
or t.Id in (
    select /*+ PRECOMPUTE_SUBQUERY */ TaskId 
    from ObjectAffectingTasks where ObjectId=12345
)