Val, check the website I am listing and see if anything matches what you're seeing.

http://yardener.com/YardenersPlantHelper/LandscapePlantFiles/VinesandIvy/Honeysuckle/ProblemsofHoneysuckle

Here is a link that might be useful: Diseases and pests of Lonicera

kayjones, thanks! Yeah, that's the site I found when I googled, but I can't tell what I'm seeing. These bugs seem too small to be aphids -- I can barely see them -- but there's no "webbing" present so I can't conclude that they're spider mites either.

Those are aphids. There are some 4000 different species of aphids and they come in a range of colors and sizes. While honeysuckles do have their own dedicated aphid species, Hyadaphis tataricae, they can be infested with a bunch of other species as well.

Blast them off with water or use an insecticidal soap.

gardengal, thanks!