Bring a large pot of salted water to a boil. Add the green beans and cook until just tender, 3 to 5 minutes.
Meanwhile, in a bowl, combine the lime juice, vinegar, cayenne pepper, salt, and tarragon. Whisking constantly, slowly add the oil in a thin, steady stream until emulsified.
Drizzle the vinaigrette over the beans before serving. Serve warm or at room temperature.
Make-Ahead Note: This dish can be partially prepared up to 2 days in advance. Blanch the green beans by boiling until tender. Plunge them into an ice bath to stop the cooking. Drain and refrigerate, covered, until ready to add the vinaigrette. Cook the green beans until heated through.