It is all relative. Certainly more expensive to make the initial build than making e.g. a road on surface, but the maintenance costs afterwards can be a lot less. Also, even those build costs can be considered acceptable if it saves enough in other costs, or brings other benefits (like less time wasted in travels, or saves space for something else). Depends on the environment, too (does it get snowy in winter, how difficult the ground is to dig, etc).
Consider that even in such "small" cities like Helsinki (or the capital area in general, still a petty village on the big world standards), they are almost continuously adding new (short) tunnels all over the place, are moving some large roads underground (either in tunnel, or digging into a pit and adding a cover to build buildings on top of), and are currently pre-planning (i.e. not decided on yet) one large tunnel complex (for cars) going pretty much past the whole main city center, and also pre-planning a huge tunnel from Helsinki to Tallinn. Last one is certainly estimated to be quite pricey, but they also estimate that it has a good chance to be worth it in the long run. Note also that pretty much all tunnels here go into hard bedrock. Some consider that as expensive, here they have done it so long that IIRC, now they like it more than softer materials (because less money is needed for making supports etc.) Ages ago they also made the main tunnel network for putting some heat/water/whatnot pipes/cables into, large enough to drive maintenance cars in. I'd think if tunnel digging was "crazy" expensive, digging tunnels for mere pipes would not have been on the top of the project list to fund.
Oh, and this one https://en.wikipedia.org/wiki/P%C3%A4ij%C3%A4nne_Water_Tunnel. A bedrock tunnel (and world's second longest tunnel, too) just for water (and a bit of electricity generation from that water flow along the way) - We must have been crazy.. 
All that said, I'm not trying to claim that Musk's this particular idea is good, I just wouldn't drop it due to just including tunnels in the idea. (He has some other, lets say "less than good" ideas, too.)
I would not disqualify this concept solely on the basis of tunnels generically being expensive. If we think about the end game being: 'get people where they want to go faster and/or lower cost'.
Important considerations for Los Angeles:
1. Seismic activity increases the cost of any structure considerably.
2. The city is spread out over a large area
3. The nature of the industries in LA avoid a concentrated rush hour - it's just busy all the time.
4. As slow as the freeways are, the trains, buses, and subways are almost always slower even if the stations are close to your endpoints.
If you have to get to the station, get on the subway, then transfer to a bus, then walk a mile.....it makes the cars look like they are traveling at the speed of sound. So, for a method of transportation to really have a positive impact - it needs to be vastly larger and vastly more efficient than the existing train, subway, and bus system already in place. Even if public transportation was free to use - I would still have a car because I cannot waste so much time. The number of tunnels and on/off stations would have to be enormous before anyone would be able to see an improvement. At that point, presumably, obscene amounts of money would have been spent either by the city or private industry. The city would need to see a productivity improvement to justify the cost in tax revenue, the private industry needs to see an operating profit. Either way - it needs to make financial sense which, at best, is a very long-term outlook measured in decades if it ever works at all.
Musk loves very long-shot ideas and perhaps he still has the charisma necessary to raise the massive capital. My thinking is that he will struggle to get anything off the ground unless Tesla planes out and trands in the direction of profitablity. After all, making a car company is a much smaller task than developing an all-new transportation system from scratch under one of the busiest cities on the planet.
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