The first suggests the speaker does not still live in Los Angeles, whereas the other three don't specifically imply whether or not the speaker still does.
To my ear, I would understand these sentences as follows:
I have lived in Los Angeles.
The speaker has lived in L.A. in the past, but currently does not. Example: "Where have you lived?" "I've lived in Los Angeles, New York, and Miami."
I have lived in Los Angeles my whole life.
The speaker has spent their whole life so far living in L.A., and may or may not still live there (if not currently, then recently). Example: "Why do you want to move to Boston?" "I've lived in Los Angeles my whole life, and am getting sick of the west coast." or "Have you been living in Boston long?" "No, I've lived in Los Angeles my whole life, and just moved here a few weeks ago."
I have been living in Los Angeles.
The speaker has been living in L.A. for a while, and may or may not still live there. Example: "I haven't seen or heard from you in a while. What have you been up to?" "I've been living in Los Angeles. It's great!" or "Welcome back to Houston! Where've you been?" "I've been living in Los Angeles, but got homesick after a couple years."
I have been living in Los Angeles my whole life.
Similar to the second example, but more likely to mean the speaker is still living in L.A.