Helium is produced from natural gas wells but with the prices for the gas where they have been for so long no one bothers to capture it - it's not worth it, it's just vented to the atmosphere and escapes.
So "running out of helium" basically means "running out of cheap helium and the price will go up until it establishes a balance with the cost of producing it from the ground instead of a government storage tank".
For as long as we get natural gas out of the ground, we will have helium.
I often wonder what it will be like for the next intelligent species/civilization that develops on earth. With all the easily accessible resources already mined will get get stuck in the steampunk age? Sure there might still some left where we could find it with advanced equipment, but not if you're just starting out.
Do planets essentially get one shot at advanced civilization?
Not sure if this is practical. In contrast to ore which is basically rock and more-or-less pure iron (or other desired metal), automobiles are a wild mix of different alloys which would have to be extensively processed (if this is possible in the first place). With plastics, the situation is similar if not worse, given that not all plastics melt, and burned fuel is burned and cannot be recovered at all.
The only thing where our civilizational waste can be used is if someone invents a Star Trek replicator clone.
Namely, the known quantity of a resource that can be extracted in an economical viable manner using current technologies.
So there are a number of variables that can affect the size of a reserve that have nothing to do with how much of the resource is known to exist.
When the reserves start getting low, you go look in some other likely place, and lo and behold, Y more years of reserves!
It seems more a way to fuel depressions than a useful way to analyze anything.
There are other resource concerns that are sometimes expressed in a form mimicking the peak-oil reasoning, but (with the exception of other fossil fuels and helium) the parallel is specious. Not, however, for the reason you cite (which applies equally well to oil), but because used platinum, phosphorus, lithium, and so on, are not destroyed; they can be mined from landfills or, in the case of phosphorus, the ocean. So intellectually sloppy advocates of conservation of these other resources imitate the well-founded peak-oil argument in a sad attempt to give their shabby arguments a veneer of respectability.
However, while Hubbert’s original reasoning for why an oil-extraction peak must exist is valid, his methods for estimating when it would happen have not held up.
For those who cannot afford, expensive things might as well not exist.
So although more precise, it deflects from the fact that the important aspect is economics.
I said this already -- the volume of the reserve is not related to the amount of oil in the ground. It's the amount of oil that can be _profitably_ extracted _at current market prices_. If the price increases, so does the reserve.
Similarly, if the oil price falls, the oil reserve shrinks, despite the amount of oil in the ground being the same.
However, controlled fusion would let us build fantastic rockets, so we could harvest all the helium we want from the outer planets.
(I'm assuming that population does not grow more than currently anticipated. I think this is fair because I later compare the number to our current Helium production which probably would scale up as well if the population were to significantly increase.)
The proton-proton chain reaction, the process that creates Helium-4 by fusing hydrogen plasma, releases 26.73 MeV of energy when creating a single helium atom out of 4 hydrogen atoms.  The molar mass of helium-4 is 4.002602 g/mol . With this, we can do a quick trip to our favorite unit-aware calculator, units(1), to find how much Helium would be produced if our hypothetical future civilization used hydrogen fusion for all its energy needs:
(1460000 TWh / year) / 26.73 MeV / avogadro * 4.002602 g/mol
= 8157132.36 kg / year
(1460000 TWh / year) / 26.73 MeV / avogadro * 4.002602 g/mol / (0.1786 g / liter)
= 45672633.63 m^3/year
> Helium production in the United States totaled 73 million cubic meters in 2014. 
That's 60% more than what would be produced by the future fusion reactors in my scenario. While it's true that I made a generous assumption by inflating the energy usage 10-fold, it appears that covering our Helium needs with fusion reactors is way more attainable than I imagined. We just need to cut back on wasting Helium a bit. The biggest "if" is, of course, if and when fusion reactors become economically viable.
Also current target fusion reactions are not based on the p-p chain. That is a much more difficult reaction to achieve. We're currently targeting D+T for first generation reactors. D+D is also very attractive because it is aneutronic and perhaps clever reactor design would allow for direct conversion of electricity. Energy in aneutronic fusion reactions is released as an acceleration of charged particles: a current. This current can be harvested through magnetic fields and thermal to electric efficiency can go up to 50% (maybe higher?). The actual details of this seem rather tricky since magnetic confinement devices are already tightly controlling the magnetic fields inside the reactor. It will be fun science and engineering for sure.
There are a few other commonly examined reactions, such as p+B11, but they're likely not going to be made viable reactions for energy production before D+T. These other reactions are all easier to achieve (in terms of Lawson criterion) than the stellar proton fusion chain.
I'll reuse your math with D+T=He-4+n and 3x energy production.
(4380000 TWh / year) / 17.59 MeV / avogadro * 4.002602 g/mol
= 41425784.21 kg / year
(4380000 TWh / year) / 17.59 MeV / avogadro * 4.002602 g/mol / (0.1786 g / liter)
= 231947280 m^3/year
1. as you acknowledged, the assumption that energy production would increase by tenfold
2. that all electrical energy produced would be made by fusion reactors
By the time we get to a state where a serious percentage of energy production was from fusion power we would likely be looking into more aggressively into other reactions and designs.
Even if we just said 30% of all energy used today was made from D+T fusion that would still be generating 10% of our current helium usage. That's a significant amount and a lot more than the 4 orders of magnitude that is oft cited online (https://www.reddit.com/r/askscience/comments/12r2s7/helium_i...). At a glance, it appears that the posts in this thread miscalculate how much fuel is necessary for a certain amount of energy. They glaze over that part of their calculations. I much prefer your dimensional analysis approach.
(.3 * 438000 TWh / year) / (17.59 MeV) / (avogadro's number 1/mol) * (4.002602 g/mol)
No, according to my source, it includes combustion.
(spoken in a high squeaky voice)