You do, and your nationality makes no difference, assuming that there is a public cemetery with available plots.
As a rule, residents of France have the right to be buried in the cemetery of the commune where they lived, where they died or where they have a family tomb.
It may be possible in other communes (especially if you lived there at some point or you have family there) but the mayor can refuse, giving reasons, for example because you have no demonstrable link with the area.
It is not generally possible to be buried on private property, though it is sometimes allowed, on permission of the regional prefect. Practical reasons also usually make it inappropriate – for example, if the property is sold a right remains for relatives to visit the grave.
It is possible to reserve a burial plot in your lifetime, and this may simplify matters for relatives. You, or your relatives if you did not do this, would talk to the mairie or, in some larger towns, the bureau des cimetières. Costs vary on the timespan, from around five years to perpetuity.