Since the primes aren't really random, it's not possible to state precisely the consequences of assuming that the primes are random in the same way that we can state precisely the consequences of the Riemann hypothesis or the axiom of choice. However, using the Cramér random model for the primes, one can "deduce" many conjectures about the primes, such as the Hardy-Littlewood prime tuples conjecture. There is a nice expository article by Chris Caldwell that explains how to use Cramér's model to deduce all kinds of number-theoretical statements.