When I am questing and/or exploring in diablo 3, I sometimes miss the New Lore popup in the bottom right corner of my screen. After a short period of time, it disappears. Can I read or listen to this lore or is it lost till my next play through ?
Yes, you can.
Press "J" to open your quest journal. Click on the Journal tab, and you can review all the lore you've collected including listening to it.
It shows the "Most Recent" by default, but you can choose whether you want it to show Quest, World, People or Bestiary lore.