[flutter_tool] Delete skipped tests (#136364)

These tests have been skipped due to flakiness for a long time.

Closes https://github.com/dart-lang/webdev/issues/1562
Closes https://github.com/flutter/flutter/issues/124214

packages/flutter_tools/test/web.shard/output_web_test.dart

@@ -5,7 +5,6 @@
 import 'package:file/file.dart';
 import 'package:flutter_tools/src/base/file_system.dart';
 import 'package:vm_service/vm_service.dart';
-import 'package:vm_service/vm_service_io.dart';
 
 import '../integration.shard/test_data/basic_project.dart';
 import '../integration.shard/test_driver.dart';
@@ -47,17 +46,6 @@ void main() {
       .having((InstanceRef o) => o.kind, 'kind', 'Bool'));
   }
 
-  Future<void> sendEvent(Map<String, Object> event) async {
-    final VmService client = await vmServiceConnectUri(
-      '${flutter.vmServiceWsUri}');
-    final Response result = await client.callServiceExtension(
-      'ext.dwds.sendEvent',
-      args: event,
-    );
-    expect(result, isA<Success>());
-    await client.dispose();
-  }
-
   testWithoutContext('flutter run outputs info messages from dwds in verbose mode', () async {
     final Future<dynamic> info = expectLater(
       flutter.stdout, emitsThrough(contains('Loaded debug metadata')));
@@ -66,25 +54,4 @@ void main() {
     await flutter.stop();
     await info;
   });
-
-  testWithoutContext('flutter run outputs warning messages from dwds in non-verbose mode', () async {
-    final Future<dynamic> warning = expectLater(
-      flutter.stderr, emitsThrough(contains('Ignoring unknown event')));
-    await start();
-    await sendEvent(<String, Object>{'type': 'DevtoolsEvent'});
-    await warning;
-  }, skip: true); // Skipping for 'https://github.com/dart-lang/webdev/issues/1562'
-
-  testWithoutContext(
-      'flutter run output skips DartUri warning messages from dwds', () async {
-    bool containsDartUriWarning = false;
-    flutter.stderr.listen((String msg) {
-      if (msg.contains('DartUri')) {
-        containsDartUriWarning = true;
-      }
-    });
-    await start();
-    await flutter.stop();
-    expect(containsDartUriWarning, isFalse);
-  }, skip: true); // Skipping for 'https://github.com/flutter/flutter/issues/124214'
 }