Is the Earth going to evolve towards Mars' fate or Venus' fate? Upon reading on this site (and many others), one can think that Mars might have supported life in a distant past (discovery of liquid water, valleys, mountains...). 

On the other hand, Venus is the perfect example of the consequences of the greenhouse effect. A planet where life is simply impossible because no heat can escape the planet's ultra-thick, CO2 filled atmosphere.

So, with a lot of exageration, one can say that both Mars and Venus could be the Earth's distant future. 

Since the Earth is actually facing global warming and greenhouse effect, I was wondering if the Earth would one day be like Venus ? Or if it is going to be like Mars, with no more magnetosphere and almost no atmosphere left ?
Or if it is not going anywhere near those two planets' fate ?
Either or neither. It's impossible to tell from the present.

If runaway climate change occurs, then yes, the conditions on Venus could be a potential analogue for the kind of environment on Earth due to the greenhouse effect.

Mars's atmosphere is assumed to have been much thicker in the past, otherwise it could not have sustained liquid water on the surface; it would have evaporated away. This water is required to explain the gorges and riverbeds we see on the Martian surface today. Researchers are still unsure where the atmosphere went, but the prime candidate is through top-loss to outer space, rather than sequestration through minerals on the surface (see here).

The likelihood of such an atmospheric escape process becoming dominant on Earth is small due to the Earth's greater mass and strong relative magnetic field, which prevents ion escape. In fact, the dominant loss process on the earth is sequestration. Some estimates put the reservoirs of sequestered carbon from original CO$_2$ at 250000 times the size of the existing atmosphere. 

The alternative is that the Earth remains in its current steady state, or close to it, far in to the future. There are many mechanisms to enable such an equilibirum, such as oceanic sequestration of C0$_2$ highlighted above. There are also more contested theories such as the Gaia hypothesis. This proposes that the global biological ecosystem on the Earth is self regulating, helping to maintain life enabling conditions on the Earth. It's pretty much a dead-cert that life doesn't exist on either the Martian or Venusian surfaces, so we can accept that if the Gaia hypoethesis is true, the Earth will never reach these conditions.