You can get rid of bounties with the thieves guild, but you cannot easily get rid of a death warrant. It seems the only ways are -
There are three quest lines that will get rid of a death warrant -
Doing the "bloodbath" quest for Larrius Varro will result in him removing any Bounty or Death Warrant.
If you are member of the Thieves Guild, completing the last quest for Sugar-Lips Habasi, Big Helende or Aengoth the Jeweler and they reward you by lifting any Bounty or Death Warrant.
If you aren't Thieves' Guild and are a male character, you can do the http://www.uesp.net/wiki/Morrowind:Ahnassi%2C_a_Special_Friend, the Khajiit thief, at the Pelagiad, Halfway Tavern. Doing her last quest makes her lift any Bounty or Death Warrant. Note: you can't join the Thieves' Guild if you have a Bounty. You have to get rid of the Bounty first, before you can join.
Librarian27 also provided a way of getting rid of this death-warrant if you are on pc -
Add these lines to the Console:
"Npc ID"->SetDisposition 80
"Npc ID"->StopCombat Player
"Npc ID"->Setfight 10
PayfineThief
SetPCCrimeLevel 0
Set mt_NewCrimeLevel to 0