Does the entire surface of the earth contain organisms?
That will depend on many things. How do you define surface? Is one meter underground still "surface"? How about a kilometer?  

Also, how large an area are we talking about? You could probably find a square millimeter in, for example, Antarctica that has no organisms. I don't think you would find a square kilometer with no organisms though. 

There are organisms (called extremophiles) that thrive in the most hostile environments we know, they can survive extreme heat, cold, acidity, salinity etc. I found this very nice list:


  
  Cold – The McMurdo Dry Valleys in Antarctica are some of the coldest, driest deserts on Earth, with average annual temperatures of
  -20oC (-4oF) and less than 10 centimeters (4 inches) of precipitation a year. Scientists have found bacteria in liquid water pockets
  embedded about twelve feet deep in “solid” lake ice. Some of these
  bacteria use chemical nutrients from particles of dirt in the ice and
  use energy from sunlight for photosynthesis.
  Hot – Large concentrations of microbes thrive in Yellowstone National Park’s Grand Prismatic Springs, a hot spring with water
  temperatures up to 90oC (188oF). Other hot springs in Yellowstone are
  extremely acidic, yet are home to many different kinds of bacteria and
  microbes. Many of these microbes use chemical nutrients in the waters
  and energy from sunlight for photosynthesis. Deep underground –
  Scientists have discovered bacteria living in ground- water 5
  kilometers below the surface in deep gold mines of the Witwatersrand
  Basin in South Africa. These microbes thrive in cavities and cracks in
  rocks. Scientists are also are investigating life within and below
  permafrost in north- ern Canada.
  Bottom of the sea – Scientists have found abundant life clustered around hydro- thermal vents on the ocean floor, including bacteria,
  mussels, clams, shrimp, and giant tubeworms that can reach ten feet in
  length. Water pouring out of the vents in the complete darkness
  thousands of feet under the surface of the sea can reach temperatures
  of 113-120oC (235-248oF). The high pressures keep the water from
  boiling. Bacteria use chemicals in the vent’s water, primarily
  hydrogen sulfide, as their energy source instead of sunlight. Other
  creatures survive by eating the bacte- ria or each other.
  High Acidity – The water in the Rio Tinto in southwestern Spain is very acidic, a result of chemical reactions between the water, and
  iron and sulfur minerals in the ground. The river has a deep red
  color, like wine, because of iron dissolved in the water. Microbes
  living in the water use chemical reac- tions with iron and sulfur
  minerals to generate the energy they need. Products from these
  metabolic reactions contribute to the low pH in the environment. Many
  algae and fungi also live in the acidic waters.
  


As far as I know, there is no evidence of life in molten lava. Apart from that though, just about every habitat you can find on the earth has been colonized by life. Ask your teacher what parts of the earth she is talking about, and report back here.

UPDATE:

MCM's very interesting answer prompted me to do some more research. It seems that life has now been found even in the Atacama (taken from Azua-Bustos
et al):


  For a long time it was thought that regions of the Atacama could not
  uphold any type of life forms. However, recent culture-independent
  methods (metagenomics, transcriptomics, in situ hybridization, etc.)
  have improved the sensitivity for life detection. Thus, microorganisms
  have been found even in the driest areas of this desert, which makes
  scientists wonder about the true limit of water availability needed to
  sustain life as we know it. In this review, we summarize the efforts
  devoted to the characterization of microbial life in the Atacama
  Desert.


So, it seems that life can exist even in the Atacama. That leaves only molten lava...

Reference

Azua-Bustos et al, Life at the dry edge: microorganisms of the Atacama Desert, FEBS Lett. 2012 Aug 31;586(18):2939-45