I have no real experience dealing with the FBI, but I have worked with the ATF. They have arson jurisdiction on Federal property. They have top notch investigators and also top lab access. Now, they have NO jurisdiction over a typical arson, just as the FBI has no jurisdiction over homicide. We can ask the ATF to assist us though, and they are usually more than willing if we have a tough nut to crack. (ie: serial arsonist) We can also ask their labs to analyze something for us without having to actually ask for agents to work with.
I took a graduate criminology class on serial murder. Our professor was a former homicide detective who helped catch the co-ed killer in Michigan during the 60s. Anyway, he told us the first day of class that the biggest myth about serial killers is that the FBI has jurisdiction. They have no jurisdiction over serial murder at all, unless, as noted, it occurs on Federal property. If a killer transports a victim across state lines, then they can get involved then as well. As has been pointed out. Typically they do not get involved unless asked, and they can either help with lab work/crime scene or actually assign agents, or both.
One thing to consider is that LAPD and the FBI have some bad blood that goes back at least a decade. They typically do not work and play well together. So to answer the question, if the locals do not want help, then the FBI will not get involved.
Brady