Significance of lipids in biological membranes...?
Membranes are built from a specific class of lipids, namely phospholipids, whose key property is that they are amphiphilic and so can self-organise to form bilayers. Not all amphiphiles do this, some prefer to adopt a micellar organisation. A bilayer composed of phospholipids produces the ideal combination of a hydrophobic barrier with a hydrophilic surface.  

Although there are clearly amino acids with hydrophobic side chains, it is difficult to conceive of a proteinaceous barrier that would reproduce the properties of a phospholipid bilayer since it would not have a continuous hydrophobic phase. 

Cellulose may be insoluble, but is full of polar groups so again would not create a hydrophobic barrier - I'm sure that ions can diffuse through a cellulose structure for example.

added later

@WYSIWYG raises an excellent point in the comments. Archeal membranes are composed of a different class of amphiphile in which the hydrocarbon tails (branched; derived from isoprenoids) are attached to glycerol by an ether linkage. It's instructive to note that the overall design principles are retained however, suggesting that the self-organising properties of these molecules was the key to the "invention" of biological membranes.