Could the current climate change events cause long-term problems in the viability of Earth to sustain life?
There is one scenario in which climate change could render the Earth inhospitable to life in general, which would be a runaway greenhouse effect.  In a runaway greenhouse effect, the Earth would get so hot that the oceans start to evaporate, adding more water vapour (powerful greenhouse gas) to the atmosphere, which will make it hotter yet, until Earth ends up like Venus.
However, the current scientific consensus is that a runaway greenhouse effect as a consequence of anthropogenic global warming is not possible.

Life in general will almost certainly go on, and most consider it unlikely that the survival of humanity itself is at risk as a direct consequence of climate change.  Many other species, however, will likely disappear, as has been widely documented in the scientific literature.  As an example, complicated ecosystems depend on coral reefs, and as things stand, there may not be any coral reefs left by the end or even the middle of the century.  This will be disastrous for significant subpopulations of humanity (I've heard in a talk last week than 400 million people depend on fish supported by ecosystems around coral reefs), but it will not by itself cause the extinction of mankind.