So, recently, my Lemon Sage has been dying, one plant at a time. The rest of my herbs (Peppermint, oregano) have been fine. Today, upon closer inspection, I noticed what appeared to be cobwebs all over the dead plants. Looking even closer on that, I noticed several tiny white bugs crawling all over the webs.

Does anyone know what bugs these might be, that attack Sage and leave webs? And should I get rid of the plant as soon as possible, or will these bugs not invade my kitchen where the herbs are growing?

Probably moths of some kind. Definitely remove the plant before they spread.

Order: Lepidoptera, Family: Eriocraniidae (http://davesgarden.com/guides/bf/b/Lepidoptera/Eriocraniidae/none/none/genus/0/)