Has an NFL player ever died on the field of play during a game?

Just one.  Sort of.

In 1971 a player had a heart attack on the field during a game.  But he was not declared dead until he was taken to a hospital.  It's hard to say whether he actually died before being taken off the field to the ambulance, but he was definitely already unconscious.

His "official time of death" was 50 minutes after the game ended.

Wide Receiver Darryl Stingley was hit hard by Raiders safely Jack Tatum on a pass play in a preseason game in 1978. His neck was compressed by the hit and he suffered a spinal cord injury and was paralyzed for life as a result. He did not die of the injury.

The other answerer talked about the NFL, but I thought I'd add, back in the early days of football, people died playing football...kind of a lot.  In fact the forward pass was developed and added to the game largely as a safety innovation to reduce deaths.

The game was very different back then, looking a bit more rugby like (rugby and football developed from the same predecessor game).  It was quite common for the entire team to link up and form a human battering ram in a 'flying wedge' formation.  These would crash into the other team with a lot of force and with players often bent over with heads down.  It doesn't take a lot of imagination to realize that the poor bastards at the peak of that wedge at probably going to have head and spinal injuries.

Yes, but not in many years

Not during the game, but Korey Stringer died of heatstroke from a Vikings preseason practice.