Does the Earth orbit the Sun's current location, or its location from 8 minutes ago? ... and how could we empirically test this?
Notable commentary:  

"... the sun isn't where it was a millisecond ago, and we are revolving around where it was. It's a little like swinging a yoyo on a somewhat elastic string around yourself as you walk around a high school track. The yoyo orbits where you were, but it stays in orbit around you. To the yoyo, you aren't really moving. An observer in the stands would say you are. Someone looking from orbit would say the whole planet on which the track is built is rotating. Etc."
  - Jake Watrous

This leads to questions about experimentation:


*

*Is this realistically testable?

*Have there been any direct physical observations to confirm or refute this?


The description of the yoyo orbit makes perfect sense, so long as the string is taut like a normal yoyo string.
But when you introduce the "somewhat elastic" property of the string, more erratic behavior begins to occur - slingshot effects, etc.
So then the question becomes:


*

*Why hasn't the Earth been flung out into interstellar space, nor collided with the Sun?


More notable commentary:

A natural model would use the center of mass frame of the solar system. The effect is minuscule: The Sun moves less than 100m in 8 minutes.
  - Qmechanic

That makes sense, external frames of reference would not help the situation.
But then, if we acknowledge that this effect exists, but is minuscule, is this effect nonetheless measurable/observable with current technology?
 A: Neither, but closer to its current location, even though gravitational information can't travel faster than the speed of light.  See my answer at https://physics.stackexchange.com/a/263244/92058.
A: 
Does the Earth orbit the Sun's current location, or its location from 8 minutes ago?

Let us examine the terms in this statement and take the simple case where only the sun and the earth exists "rotating" around it:
Orbit is a path defined in newtonian gravitation  where the gravitational potential goes like 1/r and the classical mechanics solutions are conic sections, and closed orbits can be circles or elipses. Light travels instantaneously. The system of reference is well defined, for the sun and earth, as the much greater mass of the sun makes it also  practicallythe center of mass of the system.
8 minutes ago  introduces special relativity and current location forces general relativity GR) into the problem , if one considers that gravity's effect comes through velocity of light limited gravitational forces. 
To get a correct estimate one has to start with general relativity, which does not have orbits, but four dimensional space contours.Your concerns appear in the discussion of the two body problem in general relativity and are part of the need that made the developement of GR inevitable:

if gravitational influence does propagate at a finite speed, then at all points in time a planet is attracted to a point where the Sun was some time before, and not towards the instantaneous position of the Sun. On the assumption of the classical fundamentals, Laplace had shown that if gravity would propagate at a velocity on the order of the speed of light then the solar system would be unstable, and would not exist for a long time. 

The simplistic answer  , if one looks at the GR equations , is that the intuitive 'orbit' of a Newtonian world is corrected by the space time geometry of general relativity, and if one could get an instantaneous measure ( not possible in reality) the Newtonian orbit prediction would be the one that led to the conundrum of unstable orbits.  They are stable because of General Relativity.
Thus:

Is this realistically testable?

GR predictions have not been falsified up to date, and they are tested continuously

Have there been any direct physical observations to confirm or refute this?

GR is tested every moment practically, because the GPS system takes its solutions and the special relativity solutions into account in mapping the earth. A smaller problem than earth sun, but continually validating GR

Special relativity shows that if we place a clock on a satellite and compare its recorded time to an identical clock in our rest frame on Earth, the satellite’s clock will appear to be running behind. For the GPS satellites, this difference amounts to about 7 microseconds per day. On the other hand, general relativity shows that those same clocks will tick 45 microseconds per day faster.

A: When we look at the sun are we seeing it where it was 8 minutes ago? I think this answers the question since light waves and gravity waves travel at the same speed. 
A: According to Wikipedia, the sun moves approximately at $0.0012c$ relative to the CMB. This means that in the CMB reference frame, the position of the Sun would be offset $0.0012R$ from the point that the Earth orbits according to your reasoning ($R$ being the radius of Earths orbit around the Sun). Why would this cause Earth to be flung into interstellar space or to collide with the Sun? Remember that other effects of the sun are also delayed, so Earth would not appear to, for instance, receive more sun radiation when it was on the "forward" side of the Sun.
So what if we choose a reference frame in which the Sun is traveling close to $c$? In this reference frame you would need to work out a lot of relativistic effects. Maybe someone else will supply this treatment, but we can be pretty sure that it will not show that Earth is flung into interstellar space or collides with the Sun, because that would not be compatible with the what happens in other reference frames.
A: The quote of Jake Watrous is vacuous nonsense, it should be disregarded without any consideration. 
For example, the statement "To the yoyo, you aren't really moving" is strictly false. Relative transverse motion IS motion. The yoyo and you are in relative motion, and there is an absolute quantity of angular momentum and and absolute quantity of rotational kinetic energy. (See: Newton's spinning bucket)
Furthermore, The Earth does NOT orbit around the position of the Sun 8 minutes ago, or 9 minutes ago, or 7 minutes ago, or any other random time interval suggested by a crank on the internet.
Strictly speaking, the Earth does not orbit the Sun at all. The concept of "orbit" implies that there are only two bodies being considered, in reality there are many bodies, plus interplanetary gas and dust.
An object does NOT orbit around the position of its companion object where it "was a millisecond ago", or any other random time interval suggested by a crank on the internet. 
Speaking loosely, one might say that the Earth, Sun, and planets all "orbit" around the Solar System Barycenter (SSB), but technically, that isn't a well defined statement.
A: I don't know much about physics, but if I am guessing correctly, I  think the question as currently framed, doesn't really express the doubt in questioner's mind.  Consequently, some of the answers go off on a tangent.
I think the real doubt is about the fact that the sun is not literally where we 'see' it, it was there 8 minutes ago. So this is not a question about frames of reference - the problem occurs for any frame of reference.  Nor is it a question about Yo-yos, because for a yo-yo on a 1 metre string, where we see it, is its actual position, what with light being so quick and all.
So lets reframe the question.  Lets go somewhere way out in the universe, a billion light years from any other mass, so that the gravitational effects of other bodies can be ignored. 
Say l=300000km, time t=0. We put down a marker Z with no apparent velocity, and measure everything from Z. Let's set a large mass moving away from Z in a straight line at .5l/s (at t=0 Psun=(0,0)), and 1 light second away we project a ball in orbit around that mass (at t=0, Pball=(l,0)) and start our clock. After 1 second, the sun will be at (.5l,0), but the ball will see the sun at (0,0).  
An ant sitting on the ball at t=1 wants to check its orbit.  But for the centre of the orbit should it use the (0,0) it can see, or the (.5l,0) where the sun currently is?
In other words, is the effect of a gravitational field felt instantly, or does it take time to get there?
I stand to be corrected, but it seems to me that changes to the gravitational field, as created by 2 black holes collapsing, travel at the speed of light; but the fixed curving of space caused by a single massive object, travels around with the object taking effect 'instantly' as it is already 'out there' at some distance from the object. 
So, I think the ant should use the (.5l,0) where the sun actually is, even though it its eyes are telling it the sun is at (0,0).
Now, unfortunately, our precious marker Z will have moved a bit with the gravitational field of the sun, so neither co-ordinate is actually correct, but I think the underlying doubt is clearer.
The warped space of the sun, is moving at the same speed as the sun, is it not?  The gravitational dimple that our planet is orbiting around in, is already out here. It doesn't need 8 minutes to get out here, because that distortion was already 8 light minutes away from the sun. So the gravitational effects of the sun always take effect from the sun's current position, not its apparent position.
