This is inspired by Evidence that the Solar System is expanding like the Universe?, which referenced an article by G. A. Krasinsky and V. A. Brumberg, "Secular Increase of Astronomical Unit from Analysis of the Major Planet Motions, and its Interpretation," pdf available here.
The article seeks to explain the increasing Earth-Sun separation (orbit-averaged separation, as in semi-major axis), and a large part of it is focused on the expansion of the universe. Now Ben Crowell gave a great explanation here as to why the effect of cosmic expansion only comes in with the third time-derivative of the scale factor, noting that the effect is "undetectably small."
So then I wonder: What is the rate at which the Earth-Sun distance is changing? Krasinsky and Brumberg cite something like $15\ \mathrm{cm/yr}$, but given the number of wrong calculations they do elsewhere, I don't entirely trust this claim. Who has done this analysis, and what do they find? Also, what methods are used? I would guess that measuring the precise distance to the Sun's surface would be challenging, due to all the activity it undergoes.
Note that discussion of cosmic expansion was background motivation for what sparked this question – I fully understand that any secular change in the Earth's orbit will be dominated by other effects.
Also note there is a related question on this site: How is distance between sun and earth calculated? However, that does not go into detail about the method. Moreover, it is not clear that the answers provide work for measuring small changes in the separation. For example is radar really that precise? Perhaps it is, in which case justification of that would serve as an answer here.
Best Answer
According to E. V. Petjeva (2011), the measured rate of change of the Earth-Sun distance (astronomical unit) is (1.2 +/- 3.2) cm/yr, with the uncertainty value representing 3 standard deviations. In other words, any change is within the uncertainty of the measurement. She specifically addresses the Krasinsky and Brumberg value. E. M. Standish has also addressed this issue.
The measurements are by radar echos off other planets and radio signals from space craft. See the below references for details.
(Note that "astronomical unit" is not technically the same as Earth-Sun distance, see Standish reference for details, Krasinsky and the others are all measuring the "astronomical unit"; also "astronomical unit" was redefined to be a constant in 2012, see Nature reference for details)
http://syrte.obspm.fr/jsr/journees2011/pdf/pitjeva.pdf
http://adsabs.harvard.edu/abs/2005IAUCo.196..163S
http://www.nature.com/news/the-astronomical-unit-gets-fixed-1.11416