There are scenarios where the effects of planet accretion on a sun-like star would be very significant indeed, at least in the short term. Whilst the amount of mass accreted by the star would be a tiny perturbation, the amount of accreted energy and/or angular momentum may not be.
Scenario 1: The scenario where a hot Jupiter just drops into the star from a distant radius would certainly have drastic short-term effects. But short-term here means compared with the lifetime of the star.
The kinetic energy of say Jupiter falling to the star's surface from a distant (more than a few solar radii) would be of order $GM_{\odot}M_\mathrm{Jup}/R_{\odot} \sim 4\times 10^{38}$ joules.
For comparison, 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 the kinetic energy would do work and lift the convective envelope. In other words, the star 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 star 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 star, then much less energy would be deposited in the convection zone and the effects would be lessened.
On longer timescales the star would settle back down to the main sequence, with a radius and luminosity only slightly bigger than it was before, commensurate with its tiny (0.1%) increase in mass.
This all assumes that the planet 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 star and (gas giant) planet are almost identical. So it seems likely that the planet would be starting to be tidally ripped apart, but as it is travelling towards the star 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 a Jupiter into a Sun-like star 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: A hot Jupiter arrives at the Roche limit (just above the stellar surface) having 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 the planet 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 star, since it must lose some of its angular momentum before it can be accreted.
How much of the star'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 $\sim 0.1 R_{\odot}$.
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 star will accrete $\sim 10^{42}\ \mathrm{kg\ m^2\ s^{-1}}$ of angular momentum, which is comparable to its current angular momentum of the Sun.
The accretion of an exoplanet in this way is therefore sufficient to increase the angular momentum of a star like the Sun by a significant amount. In the long term this will have a drastic effect on the magnetic activity of a Sun-like star – increasing it by a factor of a few to an order of magnitude.
