Parker Solar Probe passing extremely close to the Sun; what relativistic effects will it experience and how large will they be? note: This is a question about relativistic effects. I've included some detail about the spacecraft and its orbit for background, but the question is about relativistic effects and their observability. 

When the Parker Solar Probe passes within 8.5 solar radii of the Sun, it will be moving really fast.
A perihelion and aphelion of about 6.6 and 109 million kilometers suggests a semi-major axis of 57.8 million kilometers. The standard gravitational parameter for the Sun is:
$$GM_{Sun} \ = 1.327\times 10^{20} m^3/s^2$$
So using the vis-viva equation:
$$v^2 \ = \ GM_{Sun}\left(\frac{2}{r}-\frac{1}{a}\right)$$
the orbital velocity at perihelion will be about 195,000 m/s, or about 25 times faster than a satellite in LEO. This is close to one part per thousand of the speed of light! I think this will be a new record for a spacecraft velocity relative to the solar system.
Depth of a gravitational potential well for an object of mass $m$ is approximately given by:
$$u_G \ = \ -GM_{Sun}\frac{1}{r}$$
At 6.6 million kilometers the spacecraft will be 23 times deeper in the Sun's gravitational well than it would be at 1 AU.
Will there be any particularly unique relativistic effects detectable for the Parker Solar Probe during it's close flyby of the Sun?
Just for example, during one close flyby, how much time will its on-board clock gain or lose compared to some convenient standard solar system timescale? I know there are several different choices, "GPS time" might be one but I don't want to arbitrarily choose an inconvenient one. 

above: Parker Solar Probe cropped, from here.

above: screen shot from NASA's 
Solar Probe Plus Fact Sheet 
note: the link seems to be dead now, but there is plenty of related information available at http://solarprobe.jhuapl.edu/index.php#spacecraft

above: Illustration of the Parker Solar Probe's orbit, from here.
 A: Even missions to Phobos have to consider relativity, but that's because of the necessary sensitivity of any instrument trying to measure Phobos's gravity. So my cop-out answer is it depends on what you consider significant. I'd assume that most instruments would be measuring pretty large properties and would not have to consider relativity, but anything that needs to be super precise would have to consider it. I can't imagine the mission doesn't plan to measure relativity with some instrument.
Others are more qualified to give you numbers as to how much time will be "lost", but I know the equation for gamma, the factor by which time slows, is
$$\gamma=\frac{1}{\sqrt {1-v^2/c^2}}.$$
Therefore, traveling at 195 km/s, the probe would experience time about 0.2 parts per million slower than "normal".
Then using the equation for gravitational time dilation:
$$T=\frac{T_0}{\sqrt{1-\frac{2GM}{Rc^2}}}$$
Gravitational time dilation also slows time by about 0.2 parts per million. I'm not positive how these two interact, however. I doubt it's as simple as addition.
A: From Neil Ashby's 2006 Relativistic Effects in the Global Positioning System (or their 2003 Relativity in the Global Positioning System (Living Rev. Relativity, 6, 1 if you are ambitious) the lowest order terms to the relativistic frequency shift of a clock in orbit around a gravitational body are:
$$ \frac{\Delta f}{f} \approx -\frac{\Phi}{c^2} - \frac{v^2}{2c^2} =  -\frac{GM}{r c^2} - \frac{v^2}{2c^2},$$
where the first term is the gravitational shift and the second is time dilation. Plugging in the vis-viva equation where $a$ is the semimajor axis:
$$ v^2 = GM\left( \frac{2}{r} - \frac{1}{a} \right)$$
gives
$$ \frac{\Delta f}{f} \approx -\frac{GM}{c^2}\left(\frac{2}{r}-\frac{1}{2a} \right).$$
With a perihelion of 6.6 million km and a semimajor axis of 57.8 million km, that gives frequency shifts of 4.3E-07 (half a ppm) and 1.3E-08 at perihelion and aphelion, respectively.
See the several answers to How to calculate the planets and moons beyond Newtons's gravitational force? in Space SE for further references.
To calculate the approximate total shift in elapsed time you would have to integrate over the orbit. With a period of about 3 months (87.7 days) I get a total shift of about 0.3 seconds per orbit:
Python script: https://pastebin.com/EyJbfQVZ

A: I've talked to one of the engineers who is involved with timekeeping on the Parker Solar Probe mission. I have not yet talked with the principle timekeeping lead, so this answer may have to be revised.
There seems to be two relativistic effects that will affect PSP due to the Sun's gravity well. The first is clock error due to relativistic effects near the Sun, and the second, as mentioned by DuffBeerBaron above, is spacecraft ranging for navigation.
Of the two, the effect on ranging is the more significant. The relativistic effects on the clock seem to be very small compared to oscillator drift, which is very carefully tracked. Corrections made for oscillator drift will also correct for any relativistic effects.
The effect on ranging calculations is due to what is called the
Shapiro delay. From the paper linked below:

Shapiro  delay  is  an
increase  in  the  travel  time  (the  OWLT)  of  a  signal  passing
through  the  gravitational field of a massive body, in this case the
Sun.  It was first confirmed by Irwin Shapiro in the 1960s using
measurements of radar distances to Venus and Mercury. The Shapiro
delay is important not only to the timekeeping as discussed here, but
as a consideration in the orbit determination process itself for all
deep-space  missions.  Both  the  radiometric  range  measurements
and  the  Doppler  velocity  measurements  are markedly affected  by
the  Shapiro  delay, so this  delay  is  routinely included in
deep-space  navigation calculations.  This  effect  is  important  for
all  portions  of  these  missions;  it  is  not  limited  to  solar
conjunctions, but it is more pronounced near those events.

Note that this effect is not due to the spacecraft itself being inside the Sun's gravity well; it is due to the radio transmission through the gravity field. Because of this all deep-space missions are affected by it; it is not a unique consideration for PSP. The paper cited used data from two missions: MESSENGER, at Mercury, very close to the Sun, and New Horizons, out at Pluto, very, very far from the Sun. But both had to transmit through solar conjunctions, sending the radio signal right through the Sun's intense gravitational field.
The paper linked below describes the Shapiro delay measured by MESSENGER and New Horizons timekeeping experiments, the second is the Wikipedia article about it.
Sorry for not including any mathematical equations in my answer! That is beyond my area of expertise! I will update this answer with any further information I obtain.
Cooper et al. (2012): MESSENGER Onboard Timekeeping Accuracy
Shapiro delay From Wikipedia
