Since "philosophy" has been mentioned in this thread, I would like to point out that, depending upon where one is situated within philosophy, one might hear different answers to this question.
There is a tendency within Anglo-American schools of thought to emphasize the concept of narrative within philosophy; and to some extent this can be traced back to certain roots of philosophy which derive from ancient Greek culture wherein 'thinking' per say was likened to a "theater of the mind." It is almost as if the mind is a stage and thought proceeds with various characters stating their cases (such as happens in the Socratic dialogues of Plato). One again sees this kind of approach reinforced with the introduction of psychoanalytic theory and the insistence that yes, there is a story there that qualified individuals are able to recite even if the subject of the exercise is unaware that they are in fact caught up in this narrative.
Continental philosophy in general, and post-structuralism in particular, do not subscribe to this approach. In that school of thought, such approaches are considered derivative, representational: they do not capture the essence of thinking because they are not generative. The issue becomes much more pronounced when the experiences of WW II are factored in, with the French schools of philosophy demanding that people accept responsibility for the thoughts they think and be held accountable for their production; so, the backlash against ideology there is so strong that the idea of narrative as pre-constructed thought to be accepted (understood) without question comes to be seen as something which is bordering upon fascism.
So: yes, a photograph can tell a story but a photograph can also make a viewer think. From here, the debate rages over whether "thinking" is more properly the telling of a story or if it is instead more authentically something else: a catalyst for new thoughts forming.
If a photograph "tells a story" then everyone has more or less the same narrative in mind when they view it; but if a photograph causes thoughts to form, then different people will think of different things when viewing it.
Myself, I tend to take a slightly different approach: in comparing painting with photography, I've come to the conclusion that painting is characterized by the way marks the hands make cause the eye to trace outlines that produce ideas; but, photography is all about how we move around on our feet to capture images. Is that a story, that a photograph takes us to a different place and gives us a different point of view? Not really; stories can start there but so can new thoughts.
So my answer would be: no, a photograph does not have to tell a story and, in fact, it is of the essence of photography that it does not tell a story but rather gives different points of view from which new ideas might be thought.
Anyone with a taste for philosophy and photography can have a look at the essay I wrote about this, located here:
http://www.rhizomes.net/issue23/index.html
... first essay (in two downloadable PDF's) after the introductory article by the editor.
There is a lot of philosophy therein so don't say I didn't warn you ;-)