I read something like that years ago. I forget why, it was probably due to the size or radiation of the sun. Basically outside this window life could only exist in the oceans.
Does anyone know what that theory is called, what the window is and why terrestrial life can’t exist outside of it?
Wikipedia has a nice chart showing how the sun’s luminosity, size and temperature will vary over time. Basically, it gets bigger and brighter over lifespan
So even though the Sun will not consume Earth in a red giant for probably 5 billion years, there will be a point at which Earth is sill in existence, but nevertheless uninhabitable. 2 billion is probably a good estimate (using the linked table, that would be a 20% increase in luminosity), though I think I read someone say we might have much less - perhaps just a few hundred million years.
I don’t know that I’d count on oceans to be the answer. Once it gets hot enough, there won’t be any oceans.
Less, actually.
I don’t know about a 2 billion-year window, but I’ve seen estimates that the earth won’t be able to support terrestrial life in 0.8 to 1.2 billion years. The earliest estimates I’ve seen of land-based life was almost 500 million years ago, so maybe there’s some rounding to get the total of 2 billion you remember.
The expected cause of extinction in that timeframe is increased solar output, leading to increased weathering of silicate materials that pulls CO2 out of the atmosphere until it reaches a level where photosynthesis no longer functions. This Wiki article explains it a bit, although it’s not very well written or cited. I’m not finding something better in a brief search.
Life will keep evolving during this transition, so it’s possible that land-based life forms could continue past this point, but it’s more likely such organisms would be ocean-based. But it’s really a guess. So I’d say the answer to your question is “maybe”.
Is this at a rate that’s more or less than the amount of CO2 being released from limestone by the production of lime?
As far as the overheating, someone’s come up with a fix that involves using an asteroid to transfer orbital energy from Jupiter to the Earth. It’ll very graduallly move Earth away from the Sun fast enough to keep it cool.
All other things being equal, the increased size of the Sun increases Earth’s temperature by about 1°C every 150 million years. (If this held it would be warmer today than in the Mesozoic Age of Dinosaurs, but of course it isn’t.)
… Because other things are not equal; the Earth has been colder during the Pleistocene Ice Age than at any time since the Permian. The reason the Earth was hotter in the past was the large amount of greenhouse gases, especially CO[sub]2[/sub]. Greenhouse gases are much lower now than in the past, and will have to drop further to keep the Earth cool. Unfortunately, CO[sub]2[/sub] is essential to plant life which in turn is essential to animal life. The gas essential to advanced life is heating the planet.
So Earth’s time is running out; we’ve only a billion years or so left. Let’s not blow it! 
ETA: Or let’s hope humanity retains intelligence long enough to engineer the orbit change Dtilque mentions.
A feersum endjinn indeed!
That may break the record for earliest cite used on this board.
It’s tough to find records from the pre Usenet days, but if you dig deep enough…