Test breaks when using Future.delayed

You need to give the tester enough time to process all the async calls you have scheduled, try change your smoke test to something like this:

void main() {
    testWidgets('smoke test', (WidgetTester tester) async {
    await tester.pumpWidget(MyApp());
    // Since you wait 1 second to start the animation and another 1.4 to complete it
    await tester.pump(Duration(seconds: 3));

    expect(find.byType(MyHomePage), findsOneWidget);
  });
}

You will also need to move the Future.delayed out of the build() method, because this is causing a cyclic behaviour, every time you call setState() the build() is called again, change your state like that:

class MyHomePage extends StatefulWidget {
  MyHomePage({Key key}) : super(key: key);

  @override
  _MyHomePageState createState() => _MyHomePageState();
}

class _MyHomePageState extends State<MyHomePage> {
  bool _showFab = false;

  @override
  void initState() {
    Future.delayed(const Duration(seconds: 1), () {
      setState(() {
        _showFab = true;
      });
    });
    super.initState();
  }

  @override
  Widget build(BuildContext context) {
    return Scaffold(
      floatingActionButton: AnimatedOpacity(
        opacity: _showFab ? 1.0 : 0.0,
        duration: Duration(milliseconds: 1400),
        child: FloatingActionButton(
          onPressed: null,
          child: Icon(Icons.add),
        ),
      ),
    );
  }
}

The test framework runs with FakeAsync by default, that has some limitations and might cause the error you get.

You can explicitly run with async proper like:

void main() {
  testWidgets('smoke test', (WidgetTester tester) async {
    await tester.runAsync(() async {
      await tester.pumpWidget(MyApp());

      expect(find.byType(MyHomePage), findsOneWidget);
    });
  });
}

You might still need suggestions made in other answers.