How can I set Z up coordinate system in three.js?

You can set the up vector of the camera using

camera.up.set(0,0,1);

Then, it will work like you expect.

The answer above works in simple case, but if you wish for example to use the editor, you better set before doing anything

THREE.Object3D.DefaultUp = new THREE.Vector3(0,0,1);

So any new object will also use this convention. Using the previous answer, I struggled in the editor on on all the implications around the controls, saving the objects etc...

Please note that if you use a grid you still have to rotate it so that it covers XY plane instead of XZ

var grid = new THREE.GridHelper( 30, 30, 0x444444, 0x888888 );
grid.rotateX(Math.PI / 2);