No, it seems that low power electromagnetic fields(such as from BlueTooth devices) do not cause any significant increase in the permeability of the blood-brain barrier.
Research regarding effects the low power electromagnetic fields(EMF), such as those produced by common wireless devices, is sparse. While a few studies(as mentioned in the EMFacts article) have found that EMF cause permeability increases in the brains of lab rats, the increases tended to be very small and the tests were not very rigorous, and little(if any) research has been done regarding the effects of low-level EMF on human blood-brain barriers. More recent research, including studies specifically designed to mimic positive studies, have not shown increases of permeability of the blood-brain barrier.
This is an analysis of studies concerning low power EMF effects on the permeability of the blood-brain barrier. Unfortunately it's paywalled, but thankfully my old university credentials still work so I could get the full article.
The article's abstract gives some background on how EMF could affect the blood-brain barrier:
Exposure to levels of radiofrequency electromagnetic fields (EMF) that increase brain temperature by more than 1 °C can reversibly increase the permeability of the BBB for macromolecules.
Essentially, high frequency/power EMF can heat the barrier and increase permeability. However, this EMF of this strength is not produced by low-powered cell phones, BlueTooth devices, etc., and research regarding low power EMF effects is lacking:
The balance of experimental evidence does not support an effect of ‘non-thermal’ radiofrequency fields with microwave and mobile phone frequencies on BBB permeability. Evidence for an effect of the EMF generated by magnetic resonance imaging on permeability is conflicting and conclusions are hampered by potential confounders and simultaneous exposure to different types and frequencies of EMF.
The research mentioned in the claim is lacking: Researchers mentioned by the EMFacts article are mentioned in the analysis, and describes their findings regarding rat brains:
In a series of experiments with exposure in the 900 MHz band at different whole body SAR levels (0.02–8.3 W/kg) and pulse frequencies, a Swedish group found increased BBB permeability to albumin in the rat brain for some treatment combinations, without any obvious dose–response relation (Persson et al., 1997, preliminary findings in Salford et 86 BRAIN RESEARCH REVIEWS 65 (2010) 80 – 97 al., 1993 and Salford et al., 1994)
...
A follow-up study found a higher score for the occurrence of ‘dark neurons’ 7 weeks after a 2-hour exposure at 0.002 to 0.2 W/kg... (Salford et al., 2003).
However, the experiments had several flaws are were not very specific regarding the actual effects:
Unfortunately, analysis was qualitative only (“one larger or several leakages”=positive), there was no mention of observer blinding and exposure
duration varied from 2 to 960 min.
...
In those studies that report increased permeability following ‘non-thermal’ exposure using histological methods, no exact information is available on the number of blood vessel cross sections showing leakage.
Measured permeability increase are likely not dangerous: Even when permeability increases were reported, the actual values are quite small and inconsequential:
This would suggest a number of positive vessels per section of less than
10. The number of venules greater than 16 μm diameter per rat
brain section lies in the order of 200–300 (Park et al., 2008)... The total
number of arterial and venous cross sections of all diameters
per rat brain section would thus be estimated to exceed 500... and it is unclear why an increased permeability of less than 10 vessels per
section would have any pathophysiological significance.
Most research suggest no danger: The number of studies finding no permeability increase outweighs those that do, and the results of the research mentioned by EMFacts have not been reproduced:
Permeability studies by other research groups using mobile phone signals have generally shown no evidence for increased permeability at ‘non-thermal’ exposure levels.
...
Recently, three independent studies were published that were specifically designed to reproduce the exposure conditions of the Swedish group and used the same rat strain, the Fischer 344. The choice of this strain may be
relevant, since its reactivity to stress differs from that in other
rat strains (Shepard and Myers, 2008; Wakizono et al., 2007).
The rats were exposed in the 900 MHz band to a whole body
SAR of 0.0018 to 20 W/kg or head SAR of 0.14–2 W/kg for 30 min
or 2 h and assessed either directly following exposure or after
2 to 7 weeks. No effect on the number of albumin extravasations,
the number of ‘dark neurons’ or other neurodegenerative
markers in the brain was found (Masuda et al., 2009;
McQuade et al., 2009; Poulletier de Gannes et al., 2009).
In short, the research mentioned by the EMFacts article seems to be quite old, poorly done, and has not been reproduced using more rigorous experimentation practices. More recent research has not found anything to conclusively suggest that low power EMF, such as from BlueTooth devices, cause any dangerous increase in the permeability of the blood-brain barrier.