The gold standard for evaluating a treatment is to look at meta-analyses. This type of research paper looks at multiple studies and judges the experimental design, sample size, and other factors that may influence results to determine the overall outcome of multiple studies.
A meta-analysis for lysine supplementation as a treatment for FHV-1 was published in 2015.
Bol S, Bunnik EM. Lysine supplementation is not effective for the prevention or treatment of feline herpesvirus 1 infection in cats: a systematic review. BMC Veterinary Research. 2015;11:284. doi:10.1186/s12917-015-0594-3. link
This paper reviewed 7 studies on lysine and FHV-1 (5 on cats, 2 on cell cultures) and 10 studies on lysine and human HV-1 (7 on people, 3 on cell cultures). They concluded:
There is evidence at multiple levels that lysine supplementation is
not effective for the prevention or treatment of feline herpesvirus 1
infection in cats. Lysine does not have any antiviral properties, but
is believed to act by lowering arginine levels. However, lysine does
not antagonize arginine in cats, and evidence that low intracellular
arginine concentrations would inhibit viral replication is lacking.
Furthermore, lowering arginine levels is highly undesirable since cats
cannot synthesize this amino acid themselves. Arginine deficiency will
result in hyperammonemia, which may be fatal. In vitro studies with
feline herpesvirus 1 showed that lysine has no effect on the
replication kinetics of the virus. Finally, and most importantly,
several clinical studies with cats have shown that lysine is not
effective for the prevention or the treatment of feline herpesvirus 1
infection, and some even reported increased infection frequency and
disease severity in cats receiving lysine supplementation.
We recommend an immediate stop of lysine supplementation because of
the complete lack of any scientific evidence for its efficacy.