Would life on Earth survive without the Sun?
The answer is No, the Earth would not remain in any recognizable form without the Sun.  

There would still be a husk of a dead planet with a small amount of energy coming from the residual heat of the core which might be able to sustain chemosynthetic life in small colonies.  However, this is a pretty minor amount of energy compared to what we usually receive from the sun, and without an energy input lifeforms aren't able to sustain their unusually low entropic state (i.e. entropy wins the battle with life).  

Some other examples of everyday things which would no longer occur (some already mentioned by @fre0n) are:


There would be no seasons.
There would be no weather. 
There would be no daytime.
There would be no liquid water at the surface.
The surface temperature would drop to near absolute zero