The probiotic that I'm most familiar with is Fortiflora. Fortiflora is popular not only for it's probiotic properties, but also as a readily available source of animal digest (which can make unpalatable food smell REALLY GOOD). I have used it for this reason when getting cats to switch to wet food from a dry food diet.
I have one cat with FHV-1, and have not notice any symptom changes whenever he had access to Fortiflora.
Review of Scientific Literature
There appears to be one study on the subject (and they also used Fortiflora).
Lappin MR, Veir JK, Satyaraj E, Czarnecki-Maulden G. J Feline Med Surg. 2009 Aug;11(8):650-4. doi: 10.1016/j.jfms.2008.12.006. Epub 2009 Feb 6. Pilot study to evaluate the effect of oral supplementation of Enterococcus faecium SF68 on cats with latent feline herpesvirus
This study was reviewed in detail by the SkepVet (Dr. Brennen McKenzie):
So overall, the study provides little support for the thesis that
Fortiflora boosts general immune function in a way that would be
clinically useful for cats with FHV-1 rhinitis. ... This paper is
interesting, and the line of inquiry may lead somewhere eventually,
but it does not justify the widespread use of Fortiflora for FHV-1
Part of the reason this study is inconclusive is that there were so few animals used (12) and the results were very inconsistent (leading to no statistical differences between treatment groups).
I did not find any further studies.
Probiotics are not harmful, and can be helpful if your cat has other illnesses such as diarrhea. If you want to try it, there's no harm in it, but there's also no scientific evidence that it will help either.