It would continue to support life, but not as now. It would get half the amount of sunlight it does now and the oceans would freeze over. Even at our distance from the sun, the Earth has nearly frozen over completely several times in the geological past. So it doesn't take that much of a change to change it into a "snowball" or "slushball" planet.
Our atmosphere, though pretty good at retaining heat - still is not warm enough to keep Mars warm enough for free water on Mars. Even an Earth pressure atmosphere consisting of just CO2 is not warm enough for Mars. Even a CO2 Earth pressure atmosphere on Mars, though it might well give you open liquid water at the equator, would probably be too cold for trees.
It's a bit of a mystery how early Mars could have had oceans with shore lines, deltas etc though the evidence for this is now quite strong.
Perhaps its river systems were liquid only when the planet was closest to the sun at times when it had very elliptical orbits (its eccentricity varies a lot over timescales of hundreds of millions of years unlike Earth which has pretty much the same eccentricity all the time). And the oceans perhaps were ice covered much of the year.
Or, as an alternative, perhaps it had enough of some super strong greenhouse gas like methane to keep the oceans warm, in which case it might have had abundant liquid water just like on Earth but a rather different atmosphere rich in methane.
The water could also be kept liquid locally in places by geothermal heat with heat sources near the surface like the hot pools in Iceland. Or could flood out from such a source and then cool down once released.
But it's a bit difficult to see how you could have a lake on Mars without it freezing over. fairly quickly. Once frozen over though, it would be liquid below the surface for a long time. For instance if you had flash floods which filled a crater, the water in the crater could then be liquid for a thousand years gradually freezing over from the top. Similarly a lake on Mars could stay liquid for a thousand years or more if it was created say by an impact or a volcanic eruption which melted many cubic kilometers of ice, and heated up the rocks below perhaps creating short lived thermal heating.
However there is some life on Earth even in Antarctica, so even with the Earth frozen over in an almost snowball Earth, there would be some liquid water, just as there was for Mars. Especially if you let it have an elliptical orbit like Mars itself. But it might be mainly microbial, as well as lichens. It could have more complex life under the surface around hydrothermal vents.
And that's pretty much the type of life we are looking for on Mars. Though Mars is very cold today, and almost no atmosphere, it's still possible that it has microbial life and lichens on the surface, for instance, cyanobacteria which need bare minimum of water and survive as single species ecosystems using only a few trace elements from minerals and get nearly everything else from the CO2 in the atmosphere, what water it can get hold of, sunlight - and needs a source of nitrogen, but there are nitrates on Mars. It could also have lichens, and of course could be lifeforms that are unique to Mars that perhaps would occupy similar niches.
It also is not totally geologically inactive, in the geologically recent past it has had eruptions. None right now that we can see. But it might have geological hot spots underground which might be a source of interesting chemistry and heat. The methane plumes just possibly could point to present day life that is still existing in some habitat like that - though they could also be plumes of methane released from clathrates, created in early Mars by life but not presently - or created only by non biological causes.
And by the way, Mars gets half the sunlight of Earth, and it has a third of the gravity so needs about three times as much mass per square meter in its atmosphere to achieve the same atmospheric pressure. So that means that photosynthesis would have to work six times harder to achieve the same levels of oxygen.
If you moved Earth out to Mars, that would deal with the pressure issue for Mars as the gravity is obviously the same. So, the photosynthesis would only have to work twice as hard for the same level of oxygen. And oxygen would decline only over many centuries and thousands of years.
Still long term, if you could achieve the same vegetation levels as Earth somehow, it would have less oxygen probably in equilibrium.
But it wouldn't have as much vegetation as Earth. So with less vegetation as well because of the ice, just some cyanobacteria and lichen in patches, slow growing - and oceans frozen over, then long term it would surely lose a lot of its oxygen.
Without the plants to take CO2 out of the atmosphere and with the oceans frozen over, so that they can't absorb it either - it would gradually build up a denser CO2 atmosphere from volcanic action - which is how Earth got out of its snowball / slushball phases. But at the distance of Mars, then that wouldn't be enough to achieve a climate like our Earth.
So I expect it would stabilize with an atmosphere eventually with some oxygen but not nearly as much as we have, much more CO2 in the atmosphere than we do (which given that CO2 is poisonous for humans above 1% would make the atmosphere poisonous to us) and some plants but not nearly as much as we have now. Too cold for trees, but if it had lots of CO2 maybe have some clear water in equatorial regions when close to the sun, like "slushball Earth".
Unless of course some lifeform developed there that produced large quantities of methane which then could keep it warm enough for liquid oceans - instead of the oxygen which works well on our planet because it would otherwise be too hot - but that would be a very different planet from our present Earth.
So, Mars is very interesting, many questions to answer, but it's not a second Earth. It's in the goldilocks zone but only just, and that means life is possible there, even as now with almost no atmosphere - but it's quite hard to achieve the complexity of Earth life though it may be possible in an atmosphere with strong greenhouse gases like methane.
All this of course is assuming life as we know it. For instance one out field idea is that some form of life could be possible in supercritical CO2. If that's true, well not far below the surface of Mars it may have supercritical CO2. And all bets are off when it comes to supercritical CO2 life as we don't have any examples of it.
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