Essere is used for permanent qualities: La mappa è bianca. Stare is used for temporary conditions: La mappa sta sul tavolo. I think Duolingo has been a bit lax on this, allowing us to use essere where stare is probably better. It also applies to people: Io sono un uomo. Io sto nel letto.
EDIT: Ignore what I said! See follow-up comments below. (Now I feel stupid!)
Hmm, looks like the other comments have been deleted for some reason. I think "essere" and "stare" are a little bit like "ser" and "estar" in Spanish, but it seems to be somewhat more complicated in Italian. I still haven't worked out exactly what the distinction is. "Stare" seems to have more of a connotation of "staying" or "remaining" rather than just being a fleeting temporary condition.
But I'm no expert and have still to fully understand this myself.
Here is some information from the authority on the Italian language - L'Accademia della Crusca: http://forum.accademiadellacrusca.it/forum_12/interventi/5219.shtml.html
When talking about location, essere is used to indicate the location of something at the moment (i.e., where you can find it right now), while stare is used to indicate the place where something typically and usually stays or remains.
When talking about a person, stare is used these days to indicate someone's health, or to indicate where they live (again, the idea of where someone or something is typically found) vs. where they currently are. Stare can also be used to indicate the person's behavior or mood. However, this is a more informal way of stating things. It is also a more regional way of stating things in Southern Italy. Remember that the Italian language as an official language is still very young, relatively speaking. So, there are widespread regional differences in usage, due to the influence of the traditional language spoken in each region.
Outside of these points, there are idiomatic expressions in which one may not exchange the verbs for each other. In addition, there are verb tenses and aspects in which only one or the other may be used. For example, essere is used as the auxiliary verb for most intransitive verbs in the passato prossimo, whereas stare is used as the auxiliary verb in the present and past progressive aspects and the present prospective aspect (e.g., when one is saying that they are about to do something).
Is the English translation correct? Could it also mean "The map is above the bed. " meaning on the wall next to the bed. "On top of" sounds unusual to me. Why not simply "on the bed"? which I would expect to be in Italian "La mappa è sul letto". I know, I know, one fool can ask more than ... :)