"Actually, I don't love you anymore."
The sentence final 了, after 你, shows a change in the action or state (anymore), whereas putting the 了 right after the verb 不爱 would mean you simply didn't love the person in the past.
Actually *我不爱了你 would be wrong. Only the sentence-final 了 can with either 不, the one directly after the verb and before the object can't (the negation of that 了 is 没有+V).
And actually 爱 doesn't go with that kind of 了 either because it is a "stative verb" – a verb whose meaning is considered a fixed state rather than an action or process (知道 is another such verb). This conflicts with the function of post-verbal 了 which marks perfective aspect, i.e. an action or a process which is viewed in its entirety, from beginning to end, and fixed states have neither of those. So you can only say 去年我还爱你, but not *去年我还爱了你.
To add to that, 爱了你 is correct e.g. 他爱了你这么多年/He's loved you for so many years.
It was explained to me once that 了after the verb indicated that it happened in the past, but 了at the end of the sentence indicates that it happened in the past and is still happening. So in this case I suppose 了is replacing "anymore".