The only viral diseases for which I found this method mentioned (in the popular pres) has been Ebola (in 2014), and that's an experimental treatment.
The device, called the Hemopurifier, was attached to the dialysis machine that was already filtering the patient’s blood. The specially designed filter is made of a protein that acts as glue for proteins found on the Ebola virus’s surface. Over a period of 6.5 hours, the filter extracted the virus from the blood that flows through. While most dialysis filters can pull out molecules that are less than 4 nanometers in diameter, the virus filter boasts a mesh that’s able to filter out larger viral particles that are less than 250 nanometers. That means only the virus is pulled out, and the immune cells remain in the blood, ready to fight off any remaining viral invaders.
“We had no [idea] about how much [virus] would be extracted, because this was the first patient, but I was very surprised because the drop in viral load was deeper than I expected,” says Geiger. Before the filtration began, the patient’s virus count was about 400,000 per mL blood. After the session it had dropped to 1,000 copies/mL. [...]
Freed from that viral burden, the patient soon began to improve rapidly. His own immune system began fighting off the remaining virus, and he no longer needs dialysis or a ventilator. The patient is walking and waiting to be released from the hospital.
Geiger stresses that it’s not clear yet whether the Hemopurifier alone was responsible for the patient’s recovery, since he was given other experimental therapies, but the amount of virus removed from his body and his rapid recovery after the filtration suggests that it at least played a role in helping him survive his infection.
While puling viruses out of infected individuals has never been tried before, Geiger believes it will be an important strategy for treating not just Ebola but other vial infections as well, including HIV, hepatitis and even influenza. “It’s a very interesting concept. The big advantage is that the plasma is filtered, and only the virus is removed and the other plasma components like immune cells go back to the patient. That’s important because with viral infections, the patient is in a reduced immune situation.”
The inventors mention influenza as a possible target too, but frankly I suspect that was way optimistic.
A search in pubmed for "Hemopurifier" only finds 3 articles (and the 2001 hit is a name coincidence for an unrelated device), none of which are even about Ebola (never mind influenza), but there is (a 2009) one about hepatitis C. You can look through the (37) Google Scholar citations of that 2009 paper for subsequent research.
(The 2nd real hit in pubmed for "Hemopurifier" was a more cited 2012 article on exosome removal in cancer patients. Unfortunately, most of the citations for the 2009 hep-C paper seem to also be about exosome removal...) There is one 2014 review about Ebola though that mentions the Hemopurifier as follows:
Recent discoveries have led to the development of specific extracorporeal sorbent techniques to reduce viral particle concentrations in blood. The application of a specific device called ‘Hemopurifier' has been advocated to reduce viral load in several acute viral infections . Blood is circulated through a hollow fiber plasma filter that has a specific sorbent matrix stuffed in the compartment external to hollow fibers. Due to the fluid mechanics inside the device, plasma is produced from whole blood in the proximal part of the fibers and it is forced to flow through the sorbent bed. In the distal part of the device, plasma reenters the hollow fibers by a backfiltration mechanism. No plasma loss occurs since plasma never leaves the device (the plasma ports are left capped). According to the company's specifications (Aethlon Medical Inc., San Diego Calif., USA), the sorbent is highly specific for glycoproteins (GP) that viral envelopes. Viral particles are thereby captured and their concentration in blood is significantly reduced . This process, called ‘Lectin Affinity Plasmapheresis', is based on the concept of affinity chromatography developed in the 1970s . A unique lectin protein (Galanthus nivalis agglutinin, GNA) from Galanthus nivalis (the common snowdrop) has a high affinity to the mannose-rich GP that ubiquitously populate the surface of enveloped viruses [11,12] to inter alia mediate entry into host cells. This device has been used either alone or in series with a hemodialyzer and has displayed encouraging results both in vitro and in vivo on hepatitis C and HIV infected blood. Assuming a delicate equilibrium between the viral load and the immunological system, the eradication of a viral infection may be realized only by tipping the balance in favor of the innate immune response by lowering the circulating viral particle concentration in blood.
Unlike the Hemopurifier's invetors' own claims, this review doesn't mention respiratory viruses as a likely (useful) target. (It only mentioned Ebola, Hep C, and HIV.)
I found the (2015) scientific publication on the Eblola experiment now too; it mainly uses the generic name of the method "Lectin Affinity Plasmapheresis" (LAP), although Hemopurifier is mentioned in there. This paper is not indexed by pubmed; I think it's because it's a conference rather than a journal paper.
There's a more recent (2018) paper testing LAP (in vitro) on MERS, which is indexed in pubmed though. The fact that most of these papers appear in rather niche methods publications like the Blood Purif. journal should tell you something though.