Important papers and open problems in Boolean functions I am interested in Boolean function polynomial/rational exact/approximate/one sided approximate representation, relation to circuit/communication complexity, tools utilized to study Boolean functions (such as combinatorics, analysis, approximation theory, algebra etc).

What are some of the important references, lecture notes and key papers (including survey) in the field of Boolean functions for someone wanting to learn the growth of the area from 80s till today.

Even open problems in the area would be a good introduction as one can usually trace back the literature. One important open problem is sensitivity-block sensitivity conjecture.
Bryant's "Graph-Based Algorithms for Boolean Function Manipulation" introduced Reduced-Order Binary Decision Diagrams (ROBDDs) for representing and manipulating Boolean functions.  I believe these see a lot of use in CAD tools and circuit synthesis problems.  Bryant's paper is a good place to start a literature review for BDD.  He cites a few key earlier works and a lot of the more recent work grew out of, and cited, his paper.