Запретить прокрутку содержимого под SliverAppBar

Вам нужно SliverPersistentHeader и _SliverAppBarDelegate увидеть полный код ниже
Вы можете скопировать и вставить, запустить полный код
подробности, пожалуйста, обратитесь к этому https://medium.com/@diegoveloper/flutter-collapsing-toolbar-sliver-app-bar-14b858e87abe

фрагмент кода

SliverPersistentHeader(
              delegate: _SliverAppBarDelegate(
                TabBar(
                  labelColor: Colors.black87,
                  unselectedLabelColor: Colors.grey,
                  tabs: [
                    Tab(icon: Icon(Icons.info), text: "Tab 1"),
                    Tab(icon: Icon(Icons.lightbulb_outline), text: "Tab 2"),
                  ],
                ),
              ),
              pinned: true,
            ),

рабочая демонстрация

полный код

import 'package:flutter/material.dart';

void main() => runApp(MyApp());

class MyApp extends StatelessWidget {
  var scrollController = ScrollController();

  @override
  Widget build(BuildContext context) {
    return MaterialApp(
        title: 'Flutter Demo',
        theme: ThemeData(
          primarySwatch: Colors.blue,
        ),
        home: Home());
  }
}

class Home extends StatefulWidget {
  @override
  _HomeState createState() => _HomeState();
}

class _HomeState extends State<Home> {
  @override
  Widget build(BuildContext context) {
    return Scaffold(
        body: DefaultTabController(
      length: 2,
      child: NestedScrollView(
        headerSliverBuilder: (BuildContext context, bool innerBoxIsScrolled) {
          return [
            SliverAppBar(
              expandedHeight: 200.0,
              floating: false,
              pinned: true,
              snap: false,
              flexibleSpace: FlexibleSpaceBar(
                  centerTitle: true,
                  title: Text("Collapsing Toolbar",
                      style: TextStyle(
                        color: Colors.white,
                        fontSize: 16.0,
                      )),
                  background: Image.network(
                    "https://images.pexels.com/photos/396547/pexels-photo-396547.jpeg?auto=compress&cs=tinysrgb&h=350",
                    fit: BoxFit.cover,
                  )),
            ),
            SliverPersistentHeader(
              delegate: _SliverAppBarDelegate(
                TabBar(
                  labelColor: Colors.black87,
                  unselectedLabelColor: Colors.grey,
                  tabs: [
                    Tab(icon: Icon(Icons.info), text: "Tab 1"),
                    Tab(icon: Icon(Icons.lightbulb_outline), text: "Tab 2"),
                  ],
                ),
              ),
              pinned: true,
            ),
          ];
        },
        body: Center(
          child: Text("Sample text"),
        ),
      ),
    ));
  }
}

class _SliverAppBarDelegate extends SliverPersistentHeaderDelegate {
  _SliverAppBarDelegate(this._tabBar);

  final TabBar _tabBar;

  @override
  double get minExtent => _tabBar.preferredSize.height;
  @override
  double get maxExtent => _tabBar.preferredSize.height;

  @override
  Widget build(
      BuildContext context, double shrinkOffset, bool overlapsContent) {
    return new Container(
      child: _tabBar,
    );
  }

  @override
  bool shouldRebuild(_SliverAppBarDelegate oldDelegate) {
    return false;
  }
}