Well, there is at least one race who are stated specifically as being ageless - ala, the pad'jal. But they're not really a proper race at all as they're born normally as hyur and then are turned into pad'jal when 'chosen' by the elementals, preventing them from aging beyond late childhood. But even they're not truly immortal, they don't age sure, but they can still die through accident or illness - the Tam Tara Deepcroft clearly has pad'jali tombs within, and the WHM Job quest actually deals with recovering the ashes of a long dead pad'jal.
Other races however are stated as being clearly mortal (and in fact, the player themselves, despite being 'special' due to the Echo, still is mentioned as being mortal) - there are graveyards in the game, and in fact the death toll during the Calamity was extremely high, so much so it actually caused problems for the city states in that they quickly ran out of burial space for the deceased, hence it forced unusual means of dealing with the problem (the Church of Saint Adama Landama near Camp Drybone had to expand it's graveyard beyond it's borders just to cope with the demand, and Gridania actually resorted to reopening the formerly sealed Tam Tara Deepcroft for use as a place of internment for those lost in the Calamity).
So yes, the five races are definitely mortal.
