Biochemist here; not 100% certain how a micro-RNA survives the tea extraction process. If the claim is that honeysuckle tea has an antiviral activity, there are probably more active ingredients besides these.
http://www.nature.com/cr/journal/vaop/ncurrent/full/cr201413...
"Here, we report that MIR2911, a honeysuckle (HS)-encoded atypical microRNA, directly targets IAVs with a broad spectrum. MIR2911 is highly stable in HS decoction, and continuous drinking or gavage feeding of HS decoction leads to a significant elevation of the MIR2911 level in mouse peripheral blood and lung."
The paper does compare honeysuckle tea and synthetic microRNA, and they seem to be of roughly equal potency. The authors also tried the tea with a specific inhibitor of the microRNA, and whilst that does look it has some effect against the virus, it's very small. Sadly, the data around the question of whether the microRNA is the only active ingredient in the tea is not as strong as the data establishing that either of them are active.