As the reminder text indicates, whether the targeted land has the triggered ability that's been granted to it depends only on whether it has a blaze counter on it, not on whether Obsidian Fireheart is still on the battlefield.

The ability gained by the land triggers at the beginning of the upkeep of the land's controller, not the beginning of the upkeep of Obsidian Fireheart's controller. The player who controls the land at the time the ability triggers is the one who's dealt damage.

A land with a blaze counter on it is an illegal target for this ability. You may activate the ability targeting a land, then activate the ability in response to itself targeting the same land. Although the first ability to resolve will put a blaze counter on that land, the second ability to resolve will be countered.

If all blaze counters on a land are moved to a different land, the triggered ability doesn't follow them. The first land no longer has the ability because it no longer has a blaze counter on it. The second land doesn't have the ability because Obsidian Fireheart didn't target it.

If a land ends up with more than one blaze counter on it (thanks to Doubling Season or Gilder Bairn, for example), the ability still only causes it to deal 1 damage to its controller each turn.