Spells and abilities that would normally cause a player to gain life still resolve, but the life-gain part simply has no effect.

If a cost includes life gain (like Invigorate's alternative cost does), that cost can't be paid.

Effects that would replace gaining life with some other effect won't be able to do anything because it's impossible for players to gain life.

Effects that replace an event with gaining life (like Words of Worship's effect does) will end up replacing the event with nothing.

If an effect says to set your life total to a certain number, and that number is higher than your current life total, that effect will normally cause you to gain life equal to the difference. With Everlasting Torment on the battlefield, that part of the effect won't do anything. (If the number is lower than your current life total, the effect will work as normal.)

The "damage can't be prevented" statement overrides all forms of preventing damage, including protection abilities. Damage prevention spells and abilities can still be cast and played; they just don't do anything.

Spells and abilities that replace or redirect damage aren't affected by Everlasting Torment's second ability. They'll work as normal.

The last ability affects all damage, whether it's dealt by creatures, other permanents, spells, or cards that aren't on the battlefield. Wither works everywhere.