In Brynjolfsson's 'plasma redshift', interactions between photons and the electrons in various plasmas (solar corona, interplanetary medium, interstellar medium, inter-galactic medium, inter-cluster medium, ...) gives rise to a redshift.
This idea is but one of many that propose such an effect.
And like all its cousins, it is easy - in principle - to test, and several possible tests likely can be done by anyone reading this post*.
The first kind of test is to find a consistent 'plasma redshift' signal in one or more of the huge, freely available sets of consistent, high quality astronomical redshift data, such as
SDSS or
2dF. The 'plasma redshift' signal should be easily discernible as a relationship between the observed redshift and the expected (or modelled) integrated electron density along the line of sight to the astronomical object (this is, in Brynjolfsson's 'plasma redshift' idea, expressed in equation (1) in the fifth unpublished paper in the link Michael provided). In fact, Brynjolfsson claims just such a signal can be found in a galactic latitude correlation in just a few hundred 1a SNe, so it should be glaringly obvious in the SDSS or 2dF data (there are hundreds of thousands of galaxy redshifts in these datasets).
The second kind of test involves looking for a frequency dependence in the observed redshifts of at least some kinds of astronomical objects. Unlike doppler, gravitational, or cosmological causes of redshift - in which the observed redshift is the same across the entire electromagnetic spectrum - the observed redshift arising from Brynjolfsson's 'plasma redshift' varies with waveband**. Today, there is a lot of good data on redshifts (and line profiles) of astronomical objects observed in the x-ray, visible (and UV and NIR), microwave, and radio wavebands (there is some in the FIR too). A 'Brynjolfsson plasma redshift signal' should be somewhere in this data, though it may be harder to find than in the first kind of test.
While I don't know if anyone has done the first kind of test, starting with a clearly stated hypothesis, a great many of such tests have, in fact, been already done. For example, in rotation curve observations of nearby galaxies the 'Brynjolfsson plasma redshift signal' would likely be rather obvious (as an 'inclination correlation', perhaps). The fact that no one, apparently, has yet noticed any such signal probably tells us that it isn't there to find.
*
Given the amount of data most tests would involve, you'd best have a broadband internet connection.
**
I'm not 100% sure of this; the relevant 'theory' parts in Brynjolfsson's papers are hard to follow. Certainly the first Brynjolfsson paper clearly states there is such a relationship (see section 3).