If the cell membrane were removed from the cell, could it still survive?

The short answer is No.
The cell membrane in cells is an essential part of the cell. It separates the inside and the outside of the cell. In a way, the cell membrane defines the region that actually is the cell. It is made up of molecules called phospholipids that come together a structure called the lipid bilayer. The nature of this bilayer allows it to be selectively permeable. Moreover, various proteins are anchored in this cellular membrane, proteins that are also essential for the life of the cell. Hence, it serves as a platform for proteins and other biomolecules, and a sort of selectivity filter for what comes in and out of the cell. 
The importance of the cell membrane is hence, paramount to the survival of the cell. It maintains homeostasis, or internal balance, and also maintains the shape and structure of the cell.
By getting rid of the cell membrane, not only is the internal balance and equilibrium of the cell broken, the entire structure and shape of the cell disappears, and the cell dies.
http://hyperphysics.phy-astr.gsu.edu/hbase/Biology/celmem.html