When do prices not exist? Often we employ the use of the separating hyperplane theorem to prove existence of price vectors, when discussing infinite economies this proof is substituted for proving existence of linear functionals.
This all being said, I find it difficult to see a case in consumer theory, producer theory and general equilibrium where we cant have a price vector (unless the problem is a planners problem).
To clarify I'm asking in what mathematical environments do prices not exist and what would their economic interpretation be.
There are a bunch of examples for incomplete markets in the finance literature.  The oldest (that I know of) is Hart (1975).  In finance, the problem is that if you have two different assets they have different prices that make the market complete.  But then equilibrium considerations make them have the same price, which forces the market to no longer be complete.  (If the market was complete, then they would have the same price.)