Moo's answer is correct that, due to the separate tickets, you'll likely be out-of-luck on your ticket to Narita if your flight to LAX gets delayed. This is the biggest potential problem in my view, especially for same-day connections. I would personally allow for a long layover if doing this, especially before an expensive trans-Pacific flight that probably only operates once or twice per day. I was very glad I had an overnight layover scheduled the first time I did this, as my first flight ended up being delayed several hours due to weather. Had I not scheduled a long layover, I'd have missed my trans-Pacific flight and possibly lost the ticket, since it was a separate airline.
It's also important to note that, if you do not show up for your second flight (LAX-NRT,) the airline may cancel that entire ticket, meaning you could lose the NRT-LAX return segment, too. The airline might be nice and work with you, but they'd have no contractual or legal obligation to do so. If you see that you're likely to miss that flight, contact the airline ASAP to see what can be done.
Other issues you could run into:
If you're checking baggage, you might need to claim it in LA and then re-check it. The exception to this would be if both tickets are on the same airline or airlines that have interline agreements for baggage handling. In that case, you might be able to get them to check your bag all the way through from SFO to NRT. If you want to attempt this, please check with the airline(s) prior to booking your tickets regarding whether this is allowed for the itinerary that you're considering.
If you do end up having to claim your bag at LAX, this adds a considerable amount of time to the connection, as you'll need to leave security, wait for your bag to arrive at the claim, carry it to the check-in desk for your next flight (which will likely be in an entirely different terminal in the case of LAX,) check-in for your next flight, then clear security again before proceeding to the gate for your flight to NRT (which will likely start boarding around an hour prior to the scheduled departure time.)
If you don't have checked baggage, there's still a chance that you'll need to leave security and go to a check-in desk for the airline of your connecting flight. Whether this is required will be airline-specific. Many airlines allow online check-in these days, in which case this wouldn't be required. Some airlines may also allow you to check in from an airside airline help desk or transfer desk or at the gate. However, if the airline operating your flight to NRT doesn't allow any of these, you might end up having to leave security, go to their land-side check-in desk (probably in a different terminal,) and then enter security again.
If you can stay airside
If you don't need to leave security to claim your baggage or check in, you might be able to stay airside at LAX. There are now airside connector tunnels and bridges between Terminals 4-8 and TBIT (the international terminal) at LAX, so if you're arriving at LA on American, United, Virgin America, or Alaska Airlines, you can walk to TBIT without leaving security now.
If you arrive at LAX on Delta, there's a shuttle bus that connects the Delta gates (in Terminals 2-3) to TBIT without leaving security until they get the connector between those terminals built.
I am not sure, however, whether there is an airside connector bus between Terminal 1 and TBIT or not. So, if you fly into LAX on Southwest, you might have to leave security in order to transfer to another terminal, regardless of other factors.