What would happen if Jupiter collided with the Sun? This question is inspired by a similar one asked on Quora. Let's say a wizard magicked Jupiter into the Sun, with or without high velocity. What happens?
The Quora question has two completely opposed answers: one saying "nothing much happens" and the other saying "the Sun goes out for several hundred years". Both answers give reasons and calculations, and I know enough about physics to find both of them plausible. However ... it's plainly impossible that both answers are correct. Which one (or both?) is incorrect? Why is it incorrect?
 A: Both the quora answers are incorrect. The idea that "nothing happens" is incorrect for reasons I explain in great detail below. The idea that somehow Jupiter spreads itself across the surface of the Sun or directly influences the luminosity of the Sun by doing so is wrong on many levels as pointed out by Victor Toth on the quora page and by Rob and Chris as answers here.
Instead I put forward a couple of scenarios where the large amount of accreted energy and/or angular momentum certainly do have an effect on the Sun and/or the radiation the Earth receives from the Sun.
Scenario 1: The scenario where Jupiter just drops into the Sun from its current position would certainly have short-term effects. But short-term here means compared with the lifetime of the Sun, not hundreds of years. 
The kinetic energy of Jupiter at the Sun's surface would be of order $GM_{\odot}M_\mathrm{Jup}/R_{\odot} \sim 4\times 10^{38}$ joules.
The solar luminosity is $3.83 \times 10^{26}\ \mathrm{J/s}$.
The addition of this much energy (if it is allowed to thermalise) would potentially affect the luminosity of the Sun for timescales of tens of thousands of years. The exact effects will depend on where the energy is deposited. Compared with the binding energy of the star, the additional energy is negligible, but if the energy is dissipated in the convection zone then kinetic energy would do work and lift the convective envelope of the Sun. In other words, the Sun would both increase in luminosity and in radius. If the effects were just limited to the convective envelope, then this has a mass of around $0.02 M_{\odot}$ and so could be "lifted" by $\sim 4\times 10^{38} R_{\odot}^2/GM_{\odot}M_{\rm conv} \sim 0.05 R_{\odot}$.
So in this scenario, the Sun would both expand and become more luminous. The relevant timescale is the Kelvin-Helmholtz timescale of the convective envelope, which is of order $GM_{\odot}M_{\rm conv}/R_{\odot} L_{\odot} \sim $few $10^5$ years.
If the planet somehow survived and punched its way to the centre of the Sun, then much less energy would be deposited in the convection zone and the effects would be lessened. 
On longer timescales the Sun would settle back down to the main sequence, with a radius and luminosity only slightly bigger than it was before.
This all assumes that Jupiter can remain intact as it falls. It certainly wouldn't "evaporate" in this direct infall scenario, but would it get tidally shredded before it can disappear below the surface? The Roche limit is of order $R_{\odot} (\rho_{\odot}/\rho_{\rm Jup})^{1/3}$. But the average densities of the Sun and Jupiter are almost identical. So it seems likely that Jupiter would be starting to be tidally ripped apart, but as it is travelling towards the Sun at a few hundred km/s at this point, tidal breakup could not be achieved before it had disappeared below the surface. 
So my conclusion is that dropping Jupiter into the Sun in this scenario would be like dropping a depth charge, with a lag of order $10^{5}$ years before the full effects became apparent.
Scenario 2: Jupiter arrives at Roche limit (just above the solar surface) having mysteriously lost a large amount of angular momentum. In this case the effects may be experienced on human timescales.
In this case what will happen is Jupiter will be (quickly) shredded by the tidal field, possibly leaving a substantial core. At an orbital radius of $2 R_{\odot}$, the orbital period will be about 8 hours, the orbital speed about  $300\ \mathrm{km/s}$ and the orbital angular momentum about $10^{42}\ \mathrm{kg\ m^2\ s^{-1}}$. Assuming total destruction, much of the material will form an accretion disc around the Sun, since it must lose some of its angular momentum before it can be accreted.
How much of the Sun's light is blocked is uncertain. It mainly depends on how the material is distributed in the disk, especially the disk scale height. This in turn depends on the balance of the heating and cooling mechanisms and hence the temperature of the disk. 
Some sort of minimal estimate could be to assume the disk is planar and spread evenly between the solar surface and $2R_{\odot}$ and that it gets close to the solar photospheric temperature at $\sim 5000\ \mathrm K$. In which case the disk area is $3 \pi R_{\odot}^2$, with an "areal density" of $\sigma \sim M_{\rm Jup}/3\pi R_{\odot}^2$. 
In hydrostatic equilibrium, the scale height will be $\sim kT/g m_\mathrm H$, where $g$ is the gravitational field and $m_\mathrm H$ the mass of a hydrogen atom. The gravity (of a plane) will be $g \sim 4\pi G \sigma$. Putting in $T \sim 5000\ \mathrm K$, we get a scale height of $\sim 0.1 R_{\odot}$.
Given that Earth is in the ecliptic plane and this is where the disk will be, then a large fraction, $\gt 20\ \%$, of the sunlight reaching the Earth may be blocked. To work out if this is the case, we need to work out an optical depth of the material. For a scale height of $0.1 R_{\odot}$ and a planar geometry, then the density of the material is $\sim 3\ \mathrm{kg/m^3}$. Looking though this corresponds to a column density of $\sim 10^{10}\ \mathrm{kg/m^2}$.
For comparison, the solar photospheric density is of order $10^{-12}\ \mathrm{kg/m^3}$ and is only the upper $1000\ \mathrm{km}$ of the Sun. Given that the definition of the photosphere is where the material becomes optically thick, we can conclude that a tidally shredded Jupiter is optically thick to radiation and indeed the sunlight falling on the Earth would be very significantly reduced – whether or not the amount of radiation impacting the Earth is reduced or increased is a tricky radiative transfer problem, since if the disk were at $5000\ \mathrm K$ and optically thick it would be kicking off a lot of radiation!
How long the accretion disk would remain, I am unsure how to calculate. It depends on the assumed viscosity and temperature structure and how much mass is lost through evaporation/winds. The accreted material will have radiated away a large fraction of its gravitational potential energy, so the energetic effects will be much less severe than Scenario 1. However, the Sun will accrete $\sim 10^{42}\ \mathrm{kg\ m^2\ s^{-1}}$ of angular momentum, which is comparable to its current angular momentum.
The accretion of Jupiter in this way is therefore sufficient to increase the angular momentum of the Sun by a significant amount. In the long term this will have a drastic effect on the magnetic activity of the Sun – increasing it by a factor of a few to an order of magnitude.
A: I would definitely lean towards "nothing" happening.
The "goes dark for 200 years" answer makes an awful lot of assumptions, some of which seem unfounded to me. In particular, it assumes that Jupiter will evenly spread over the surface of the sun, and will remain on top without mixing with the bulk of the sun.
At one other extreme, if Jupiter does not spread at all, it can at most blot out around $.3\%$ of the sun's luminosity (as that is the fraction it would cover). They also mention that the mass would make the sun last longer, which is suspect, since extra mass typically reduces the lifetime of main sequence stars.
Given that sunspots larger than Jupiter form occasionally, and are quickly mixed back into the sun, it seems likely Jupiter would do the same thing. And if it is mixed through the convection layer, all it would do is marginally decrease the temperature, and thus marginally dim the sun.
A: The main difference between the "nothing much happens" answer and the "centuries of darkness" answer is that 
the first states, without an explicit calculation, that 

after Jupiter gets broken down by tidal forces, its cold material will mix up with the convective flows quite rapidly

while the second makes some thermodynamic assumptions and estimates that this "quite rapid" timescale is a few hundred years of absorbing the Sun's entire energy output, or a few thousand years of absorbing a major chunk of the Sun's energy output.
These aren't inconsistent with each other: the Sun's total time on the main sequence will be about ten billion years, and a blip of a few hundred or a few thousand years is nothing compared to that.  The Sun would be fine.
However, there's a profound flaw in the "centuries of darkness" answer.
It assumes, reasonably, that Jupiter is made entirely of hydrogen;
however it assumes an "average temperature" for this hydrogen of about 100 kelvin.  That's a reasonable estimate for the average temperature at Jupiter's cloudtops.  But given that Earth's core temperature is roughly 6000 K, it's absurd to assume that Jupiter is cold all the way through.
A quick search finds one estimate of Jupiter's core temperature of 24,000 kelvin, and an estimate that nearly all of Jupiter's volume is hotter than the Sun's surface, with a temperature of 6000 K only a few thousand kilometers below the cloudtops.
So the assumption that spreading Jupiter's mass over the surface of the Sun would turn the Sun cold for any length of time is just wrong; Jupiter's mass is mostly already hot.
There are many possibly interesting effects that I'm not prepared to model.
(For example, Rob Jeffries is currently writing a neat answer about an accretion disk; you can always tell when you've found a real astronomer because they answer every question by talking about angular momentum.)
A: Simply calculating the amount of heat generated, and comparing it to the heat capacity of Jupiter (even ignoring, as @rob says, that Jupiter's core temperature is hotter than the surface of the sun) is fallacious reasoning. What mechanism would cause the entirety of the sun's energy output be directed towards heating up Jupiter? I guess if Jupiter were spread completely across the sun's surface, that would cause a layer of mass that would have to be heated up before we would see any solar energy, but that would require Jupiter to somehow spread laterally but not radially. If Jupiter were mixed throughout the sun, the temperature of the sun would decrease slightly, and perhaps it would take a few hundred years for the sun's temperature to return to its previous level, and maybe we would get a few basis points less solar radiation, but it wouldn't go out.
