Bernoulli equation in open vs. closed systems

In a closed system where there is no flow and thus velocity = 0, Bernoulli's equation is reduced to $$ P_1 + rho gz_1 = P_2 + rho g z_2$$ $$ P_2= P1 - rho g Delta z$$ which is really just the normal rule for pressure in liquids. In gases usually $rho$ is too small so the pressure is constant throughout the gas. So basically it is reduced to the normal rules you use for fluid statics.

I suppose the OP thinks of a closed circuit with flow, provided by a pump.

This type of circuit exists for example in a PWR primary circuit, where liquid water is heated by the nuclear fuel in the reactor pressure vessel and continuously pumped to a heat exchanger, called steam generator, where a secundary (also closed) circuit delivers the steam to the turbines.

The Bernoulli equation is valid because it is a steady state flow. The presure at the small pipes of the heat exchanger is smaller than at the large entry chamber for example.