Have Einstein's field equations ever been solved for Earth? If not, what remains unknown that prevents a solution?
 A: Actually, not only the equations have been solved but also the solutions are being used every day by the millions who use the GPS system. Of course the solutions are not analytic solutions but using numerical methods to fit the time differences from the satellites controlling the GPS system and mapping the surface of the earth.
A google search of "GPS general relativity" brings a number of entries that explain what is done. Here is one
extract:

Because an observer on the ground sees the satellites in motion relative to them, Special Relativity predicts that we should see their clocks ticking more slowly (see the Special Relativity lecture). Special Relativity predicts that the on-board atomic clocks on the satellites should fall behind clocks on the ground by about 7 microseconds per day because of the slower ticking rate due to the time dilation effect of their relative motion


Further, the satellites are in orbits high above the Earth, where the curvature of spacetime due to the Earth's mass is less than it is at the Earth's surface. A prediction of General Relativity is that clocks closer to a massive object will seem to tick more slowly than those located further away (see the Black Holes lecture). As such, when viewed from the surface of the Earth, the clocks on the satellites appear to be ticking faster than identical clocks on the ground. A calculation using General Relativity predicts that the clocks in each GPS satellite should get ahead of ground-based clocks by 45 microseconds per day.


The combination of these two relativistic effects means that the clocks on-board each satellite should tick faster than identical clocks on the ground by about 38 microseconds per day (45-7=38)! This sounds small, but the high-precision required of the GPS system requires nanosecond accuracy, and 38 microseconds is 38,000 nanoseconds. If these effects were not properly taken into account, a navigational fix based on the GPS constellation would be false after only 2 minutes, and errors in global positions would continue to accumulate at a rate of about 10 kilometers each day! The whole system would be utterly worthless for navigation in a very short time.

A: We do not even have an exact "solution" for the Newtonian gravitational field of Earth.  The closest we have is the based on the Preliminary Reference Earth Model (PREM).
We do not know the exact distribution of mass or other properties of the material inside the Earth.  Even if we did it would be only be possible to make numerical calculations and not "solve" the EFE equations - there are 16 equations in the EFE even if it only looks like one.
Earth's is not even uniform in any sense.  There's a "lumpiness" to the gravitational field even at Newtonian level.
Doing so is hardly worth the effort involved for three reasons :

*

*The differences from Newtonian gravity would be extremely small.

*The existing approiximate solutions (as commeneted by @g-smith) for GR provide plenty of accuracy and more won't be useful because the local field varies so much.

*The existing "approximate" GR solutions provide enough accuracy to check experimental values (like gravitational time dilation for Earth).  More we accuracy cannot make use of for Earth.

So "solving" the EFEs for Earth is not practical to do or of any practical benefit.
