I have read that the Sun produces $2 \times 10^{38}$ Neutrinos per second weighing in at approximately 8 MeV. I have 2 questions.
Is there any way to calculate how many neutrinos have been produced in the past 13.5B years?
Is this mass part of the calculation of the "visible" mass of the universe? Or is it part of the dark matter?
The neutrino mass, as you have defined it, will not be part of any calculation of the visible mass of the universe. There will indeed be a very large number of $>$MeV neutrinos whizzing about that have been emitted by stars during their nuclear burning lifetimes.
These neutrinos would be a very small contributor to the dark matter in the present day universe, but of course would have been absent during the epoch when the cosmic microwave background was formed (which is one of the main means by which the dark matter contribution is estimated).
To get an answer to this question you would need a library of stellar models that gave the neutrino luminosity and energy spectrum as a function of stellar age. You would integrate these over the corresponding stellar lifetimes and a stellar mass distribution. A complication would be that you have to introduce a star formation rate (for the universe) and integrate only up to the present day for those stars (the majority) that have been born, but are still alive.
But we can argue that the neutrino contribution must be small because we know that only the tiny minority of a star's rest mass is converted into energetic neutrinos (on the order of 1%) during hydrogen burning. But stars (past and present) do not even contribute to the majority of baryonic mass in the universe and the majority of baryonic matter (perhaps 90%) is not and has never been inside a star.
Another way of seeing this is that only about 1% of the matter in the Milky Way interstellar medium is in the form of heavy elements and that the helium abundance has only increased from about 24% to 26% since the Big Bang. This indicates that very little primordial material has been fully processed by stars, even in our galaxy, although there is an additional contribution from those stars that have been born and are still alive. Given that converting hydrogen to helium is about 1% efficient in terms of rest-mass converted to energy, and that only about 1% of this energy is in the form of neutrinos then this looks like a very small contribution indeed.
A lot more neutrinos may be emitted by the gravitational collapse of the cores of massive stars. Up to $10^{46}$ J of neutrinos may be emitted per supernova, but this is still less than 1% of the original rest-mass energy of the star.
Given that there is about 5 times as much dark matter as baryonic matter, then hot neutrinos can make very little contribution to the former.
Daniel Darabos reminded me of Fukugita & Peebles (2004) - "The Cosmic Energy Inventory". This paper estimates (along the lines I outlined above) that the fraction of the critical density (where cosmic microwave background measurements suggest that the energy density of the universe is indeed very close to this critical value) present in stars and stellar remnants of all kinds is just 0.0025 (just 6% of the total baryonic mass) and that stellar neutrinos account for a fraction of just $3\times 10^{-6}$, dominated by neutrinos from core-collapse supernovae. This fraction is comparable to the total energy density of "starlight" but much less than the contribution from "primordial neutrinos" which decoupled from the rest of the universe after about 1 second post big-bang ($\Omega_\nu \sim 10^{-3}$ for primordial neutrinos).
A further point made by Fukugita & Peebles, that I had forgotten, is that you have to reduce the stellar-produced neutrino energy density by some factor that accounts for the expansion of the universe since the neutrinos were emitted.
To add to ProfRob's excellent answer, an excerpt from Simon D.M. White's 2018 essay, Reconstructing the Universe in a computer: physical understanding in the digital age:
A possible solution, that dark matter might be made of neutrinos, was greatly encouraged by a 1980 tritium decay experiment which claimed an electron neutrino mass of 30 eV. A critical question was then whether the growth of structure in a neutrino-dominated universe could be consistent with the large-scale structure seen in the present-day galaxy distribution.
[Neutrino dominated universes are predicted to have more extensive voids than observed.]
This discrepancy led to the abandoning of the known neutrinos as potential dark matter candidates, even though it would be another two decades before they were finally excluded by experimental upper limits on their masses.
A recent experimental upper limit is 1.1 eV from An improved upper limit on the neutrino mass from a direct kinematic method by KATRIN, 2019.
This article
confirms the number of neutrinos in your questions and says that they carry away energy from the sun at a rate of $4\times 10^{24}$ Watts.
Stars are said to form 180 million years after the big bang. Assuming they are approximately Sun-like, there would be at most $\frac{10^{53}}{10^{30}} = 10^{23}$ of them (from the mass of the universe divided by the mass of the Sun).
An estimate of the number of neutrinos formed is
$13.5 \times 10^{9}\times 365 \times 24\times 3600 \times 2\times 10^{38} \times 10^{23} = 9\times 10^{78}$
about $10^{79}$.
About $10^{48}$ kg. These figures are very rough estimates, but show that the mass (dark mass) of neutrinos would be small compared to the mass of the universe, that's about $10^{53}$ kg.
