Under unknown circumstances, a minion buffed by Hobgoblin or Mistcaller can become immortal at 0 Health. This is not a visual bug - the Server will send you actions involving it, you can attempt to attack with it and fail (it moves a bit then the Server sends you your next set of actions, and it is still selectable), it takes up space on your board, you can target it with spells and both you and your opponent see it.
It can be targeted and buffed by spells, damaged, set TO_BE_DESTROYED without dying, healed, attempted to attack or attack with (the attack fails if either minion is at 0 Health, but buffs and heals can be used to make the minion valid for combat again). If its Health goes below 0, it is set back to 0. If it has Taunt, it can't be attacked past. If it is bounced to the hand and played again, it stops being immortal. If it has a triggered effect, it works normally ( https://www.reddit.com/r/hearthstone/comments/3ruu99/vitality_totem_glitchmight_be_gamebreaking/ ).
The minions it has been observed with are Echoing Ooze, Haunted Creeper, Micro Machine, Novice Engineer and Argent Squire (and possibly more). However, an Echoing Ooze is usually (always?) on the board at some point in the past.
NEW: Reliable reproduction steps!!!
http://us.battle.net/hearthstone/en/forum/topic/19648775642?page=1#4
In addition, Blessing of Wisdom is confirmed to draw a card.
NEW: Full game footage, including the turn the Hobgoblin/Echoing Ooze is played, how it died, attempts to attack, showing that it can be revived, showing that if it goes below 0 Health it jumps to 0 Health immediately afterwards, etc: https://www.reddit.com/r/hearthstone/comments/3iiiez/twas_the_night_before_tgt_and_an_interesting/
NEW: 'The one missing picture is that the Haunted Creeper died immediately upon being replayed after the Sap.' This is after Shadowflame was played on it, so TO_BE_DESTROYED was maintained on the minion when it got sapped, when it got replayed, and it lost its immortality and died.
https://www.reddit.com/r/hearthstone/comments/3jb5va/this_haunted_creeper_really_didnt_want_to_die/
INCLUDES LOG: https://www.dropbox.com/s/t4a9gxm8q9k5o55/zerohealthcreeper.log?dl=0 (haunted creeper id=10)
INCLUDES LOG: https://www.reddit.com/r/hearthstone/comments/3izdfz/weird_brawl_bug/ -> https://gist.githubusercontent.com/jleclanche/bc19d470e132bbb4f7c9/raw/db99d55473125b03e47e545f6e8ba3267362887f/gistfile1.txt (argent squire id=7)
NEW: Apparently, very rarely, you can get an immortal weapon that does not get destroyed by negative durability: https://www.reddit.com/r/hearthstone/comments/3nrfho/1_durability_weapon/
https://www.reddit.com/r/hearthstone/comments/3nu0hi/i_recreated_the_1_durability_weapon_bug/
https://www.reddit.com/r/hearthstone/comments/3nxg75/i_also_recreated_the_1_durability_bug_with/
https://www.reddit.com/r/hearthstone/comments/3r2dte/infinite_weapon_bug_with_negative_durability/ (claims no hobgoblin or echoing ooze was played)
NEW: The bug is claimed to have happened on a Stampeding Kodo without an Echoing Ooze/Hobgoblin having been played at any point in the game. (But Faceless was, and Mirror Entity might have been.) https://www.reddit.com/r/hearthstone/comments/3nypv4/my_minions_just_stayed_on_the_board_with_0_health/
Includes video:
http://us.battle.net/hearthstone/en/forum/topic/18596063143
https://www.reddit.com/r/hearthstone/comments/3hv9ud/weird_bug_immortal_haunted_creeper/
https://www.reddit.com/r/hearthstone/comments/3isjj0/the_undying_echoing_ooze/
https://www.reddit.com/r/hearthstone/comments/3j3gtt/echoing_ooze_bug_feels_like_hacking_this_should/
https://www.reddit.com/r/hearthstone/comments/3qrlvk/funny_bug_with_echoing_ooze_turning_invulnerable/
https://www.reddit.com/r/hearthstone/comments/3r3vok/we_got_the_echoing_ooze_bug_in_the_middle_of_our/
Death caused by Power Overwhelming:
http://us.battle.net/hearthstone/en/forum/topic/18719273428
Other reports:
http://us.battle.net/hearthstone/en/forum/topic/18710642906
http://us.battle.net/hearthstone/en/forum/topic/18706412646
http://us.battle.net/hearthstone/en/forum/topic/18596063381
https://www.reddit.com/r/hearthstone/comments/3ht505/the_soul_of_an_echoing_ooze_lets_me_down_in_my/
http://us.battle.net/hearthstone/en/forum/topic/18716983597
https://www.reddit.com/r/hearthstone/comments/3i9dgp/zero_health_echoing_ooze_lives_on/
http://us.battle.net/hearthstone/en/forum/topic/18596067249
https://www.reddit.com/r/hearthstone/comments/3k3t6k/crazy_ooze_bug_in_ranked_play_shown_as_comic_strip/
It can also happen with Mistcaller instead of Hobgoblin:
http://us.battle.net/hearthstone/en/forum/topic/18705073115
http://us.battle.net/hearthstone/en/forum/topic/18710643668
https://www.facebook.com/photo.php?fbid=1140029286010922&set=gm.391801437661682&type=1&theater < seemingly has no echoing oozes, but does have mistcaller
It can also happen with Execute:
http://us.battle.net/hearthstone/en/forum/topic/18704213230
Can it happen with NO attached enchantments?
https://www.reddit.com/r/hearthstone/comments/3icezq/the_silver_hand_recruit_that_wouldnt_die/
If the immortal minion has Taunt, it prevents the opponent from attacking past it:
http://us.battle.net/hearthstone/en/forum/topic/18695102746
Lowly Squire: http://us.battle.net/hearthstone/en/forum/topic/19288909472
Lightspawn: http://us.battle.net/hearthstone/en/forum/topic/19288659907
Elven Archer: http://us.battle.net/hearthstone/en/forum/topic/19288250165
Haunted Creeper: https://www.reddit.com/r/hearthstone/comments/3per21/as_if_haunted_creeper_wasnt_annoying_enough/