You can use hate in expressions such as 'I hate to trouble you' or 'I hate to bother you' when you are apologizing to someone for interrupting them or asking them to do something.
You can use hate in expressions such as 'I hate to say it' or 'I hate to tell you' when you want to express regret about what you are about to say, because you think it is unpleasant or should not be the case.