For consideration.

These warnings are already covered by renderers based on react-reconciler. Maybe we don't gain that much by mirroring them in the shallow renderer? Maybe it's not worth the added complexity? Particularly with more complicated warnings like the ones being added for #12419 and #12404.
constructor(props, context) {
        super(props, context);
        this.state = {
          hasUpdatedState: false,
        };
      }
      shouldComponentUpdate() {
        return mockFn();
      }
      render() {
        return <div>{this.state.hasUpdatedState}</div>;
      }
    }
    const shallowRenderer = createRenderer();
    shallowRenderer.render(<Component />);
    const mountedInstance = shallowRenderer.getMountedInstance();
    mountedInstance.setState({hasUpdatedState: true}, () => {
      expect(mockFn).toBeCalled();
      expect(mountedInstance.state.hasUpdatedState).toBe(true);
      done();
    });
  });
  it('throws usefully when rendering badly-typed elements', () => {
    const shallowRenderer = createRenderer();
    const renderAndVerifyWarningAndError = (Component, typeString) => {
      expect(() => {
        expect(() => shallowRenderer.render(<Component />)).toWarnDev(
          'React.createElement: type is invalid -- expected a string ' +