What are the penalties for dying in The Division?

PVE: Nearly nothing. You will restart in a safe house. If on a Story Mission, you will restart at the last checkpoint. The npcs are also reset so you have to face the same opponents again.

Non Rogue: You will lose all unsaved Dark Zone loot items, some Dark Zone credits (~1700), a few rank Points (~1500-2000) and keys (normally i use just a single key, even if i carry ~40). Killing 10-15 mobs will get you to the same DZ credit/rank level again.

Rogue: Same as non-rogue, but a higher amount of rank and dark zone credits. After Patch 1.02: The lost amount of rank and DZ credits was dramatically decreased, it doesnt hurt too much, even if on manhunt now.

If you are rogue, your rogue timer will pause if you are attacking other players. That also applies to the 5 min counter when in Man hunt mode.

Losing Dark Zone ranking can mean a dark zone level downgrade!

Regarding DZ amounts, these appear to be a function of your current DZ rank.

For example, I died at dz rank 14 and lost ~400 dz xp. But then died again at 20 dz rank and lost ~700 dz xp. I've put ~ because the amount seemed to vary depending on how far I was through that particular dz rank.

I believe the exact numbers are about 2% of your total rank for non rogue in PvP and then the rogue level increases that. This table seems accurate compared to what I've seen:

https://www.reddit.com/r/thedivision/comments/4afdng/full_dark_zone_xp_table_and_what_you_exactly_risk/