Which human cells have the longest and shortest lifespan?
If we define lifespan of a human cell to mean the period during which the cell survives with the same function in a human then..

LONGEST (LIFETIME) 

Stem cells. Many types of stem cell form during embryogenesis and persist with the same functionality through to death. would be the longest lived (e.g stem cells that form blood cells such as lymphocytes, red blood cells and platelets).  Stem cells are found in many tissues including self-renewing ones such as the blood/bone marrow, gut epithelium and skin.

VERY LONG (YEARS-DECADES)

Neurons are long lived and are generally non-mitotic.  However, recent studies are raising questions about the degree of cell division in the 'mature neurons' of the brain.

Memory T-cells and memory B-cell (which can be activated and secrete antibodies as 'plasma cells') can provide long-lasting immunity from exposure to antigens in the environment and vaccines.

LONG (FEWER YEARS-DECADES)

Oocytes (eggs)..which are present at birth but which become functionally mature in waves  during puberty.

RACE FOR THE SHORTEST

Labeling studies show that mature gastric epithelial cells live for a few days. 

The top layer of skin epithelium is sloughed off every few days.  

Sperm cells live for days and are expended or re-cycled.

Mature neutrophils have a circulating half-life of around 8 hours but they can persist as mature forms within the bone marrow for a few days before entering the circulation and they may exit the circulation into depots such as the lung.  So the 8 hours is too short for their actual lifespan.