What plants are sensitive to the juglone produced by black walnuts?

When plants produce a chemical that prevents the growth of other plants, it is called allelopathy. 
Black walnut is a common woodland and landscape tree in the Midwest that produces an allelopathic compound called juglone which is presumed by some to cause an allelopathic response (inhibition of growth) in other plants.

Very little research has been done in native or garden soils that show that juglone has any consistent impact on the growth of other garden plants.  Some plants have been observed to grow poorly near walnuts.  This includes members of the tomato family (tomato, potato, eggplant, pepper) as well as some shrubs like lilac, hydrangea, rhododendron, viburnum, and yew.  There are also many plants that have been observed to coexist nicely underneath walnuts without any injury including shade-loving perennials like bugleweed and bleeding heart.  

Plants presumed to be sensitive to the allelopathic chemical juglone often yellow, grow poorly or look stunted, and eventually die when grown near walnuts. These symptoms can also be caused by other conditions like dry soil conditions and low light (both common under the shade of any tree, including black walnut).  

An extensive literature review and analysis has been done on juglone and its impact on garden plants.  This report from Washington State University provides these recommendations for growing any plant under a walnut tree (both those plant species thought of as "juglone sensitive" and those that are "juglone resistant"):

• Provide adequate irrigation for landscape plants during drier, warmer weather. All plant roots compete for water, nutrients, and oxygen. The denser your plantings, the more intense competition will be.
• Plant sun-loving plants in full sun—not under tree canopies. Shade-tolerant species are the best choice under walnut trees.
• Mulch well with arborist wood chips to retain soil moisture and nourish beneficial soil life including mycorrhizae.
• Enjoy your walnut trees! Not only are they robust landscape plants they provide food and habitat for wildlife. In areas with thousand-cankers disease, however, they should not be planted.
• Use walnut wood chips for mulch if you have them. They will not harm plants and work just as well as those from any other woody species.

Learn more about juglone and its limited impact on garden plants in this article: Do black walnut trees have allelopathic effects on other plants?