That's a surprisingly complicated question. If you ask the U.S. Food and Drug Administration, the answer is "yes:" a food containing coconuts is required to be labeled "contains tree nuts" under FALCPA.
The unasked question here is whether coconut is a dangerous food for those with tree nut allergies. And the answer here is, "It depends." Allergies to coconuts are believed to be far less common than allergies to, for instance, cashews and almonds (two particularly allergenic tree nuts).
Botanically, coconuts are most closely related to other palms and betel nuts. They come from coconut palm trees and are not closely related to most other tree nuts. While botanical relationships are not the only factor that determines whether two foods will be cross-reactive, foods that are close biological relatives often share related allergenic proteins. (Good example of this phenomenon are cashews and pistachios - two closely related plants that contain similar proteins. People who are allergic to one of these nuts are often allergic to both.) There is some evidence of cross-reactivity between coconuts and hazelnuts and between coconuts and walnuts.
What does this mean about coconut's role in your diet if you've been diagnosed with an allergy to another tree nut? First, you should avoid coconuts after a positive allergy test to another tree nut given that there is some possibility of cross-reactivity and that tree nut reactions can be severe. Second, if you're allergic to another tree nut, have a history of eating coconuts without problem, and are interested in seeing whether coconuts could be part of your diet, discuss the possibility of further testing with your allergist. Your allergist can let you know whether your other test results and history make more testing or a food challenge a reasonable next step.
Sources: Rieger, Mark. University of Georgia. "Coconut." Internet Resource. 21 April 2009. Teuber, Susan M. and W. Rich Peterson. "Systematic Allergic Reaction to Coconut (Cocus Nucifera) in 2 Subjects with Hypersensitivity to Tree Nut and Demonstration of Cross-Reactivity to Legumin-Like Seed Storage Proteins: New Coconut and Walnut Food Allergens." Journal of Allergy and Clinical Immunology. June 1999. 103(6): 1180-85.
