It's a yes-and-no thing. In general, yes they do. But it depends on how you store them and the expiration date is not a good measure.
 If your spices are based around essential oils, over time, especially in warm and lit environments they'll oxidize and go rancid ... and start tasting bad
 Some spices simply loose most of their flavor over time and go 'stale' ... ground pepper is one of these. Cumin another, and nutmeg, etc.
 Plenty of 'dried' spices that are not 'ground' will last a LONG time if you keep them (a) cool, (b) in the dark, and (c) well-sealed. See for example: whole cloves, peppercorns, nutmeg, allspice. You can also freeze whole dried spices that are seeds that need to be ground.
 I agree with the 'sniff test' already proposed.
"Habe nun, ach! Philosophie, Juristerei und Medizin, Und leider auch Theologie Durchaus studiert ..." (Goethe, "Faust")
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