diff --git a/cosmos/operators/local.py b/cosmos/operators/local.py
index aaad4e259..52428753e 100644
--- a/cosmos/operators/local.py
+++ b/cosmos/operators/local.py
@@ -40,9 +40,7 @@
     FullOutputSubprocessHook,
     FullOutputSubprocessResult,
 )
-from cosmos.dbt.parser.output import (
-    extract_log_issues,
-)
+from cosmos.dbt.parser.output import extract_log_issues, parse_output
 
 logger = get_logger(__name__)
 
@@ -276,7 +274,7 @@ def calculate_openlineage_events_completes(
         try:
             events = openlineage_processor.parse()
             self.openlineage_events_completes = events.completes
-        except (FileNotFoundError, NotImplementedError):
+        except (FileNotFoundError, NotImplementedError, ValueError):
             logger.debug("Unable to parse OpenLineage events", stack_info=True)
 
     def get_datasets(self, source: Literal["inputs", "outputs"]) -> list[Dataset]:
@@ -349,11 +347,12 @@ def get_openlineage_facets_on_complete(self, task_instance: TaskInstance) -> Ope
             job_facets=job_facets,
         )
 
-    def build_and_run_cmd(self, context: Context, cmd_flags: list[str] | None = None) -> None:
+    def build_and_run_cmd(self, context: Context, cmd_flags: list[str] | None = None) -> FullOutputSubprocessResult:
         dbt_cmd, env = self.build_cmd(context=context, cmd_flags=cmd_flags)
         dbt_cmd = dbt_cmd or []
         result = self.run_command(cmd=dbt_cmd, env=env, context=context)
         logger.info(result.output)
+        return result
 
     def execute(self, context: Context) -> None:
         self.build_and_run_cmd(context=context)
@@ -460,20 +459,6 @@ def __init__(
         self.base_cmd = ["test"]
         self.on_warning_callback = on_warning_callback
 
-    def _should_run_tests(
-        self,
-        result: FullOutputSubprocessResult,
-        no_tests_message: str = "Nothing to do",
-    ) -> bool:
-        """
-        Check if any tests are defined to run in the DAG. If tests are defined
-        and on_warning_callback is set, then function returns True.
-
-        :param result: The output from the build and run command.
-        """
-
-        return self.on_warning_callback is not None and no_tests_message not in result.output
-
     def _handle_warnings(self, result: FullOutputSubprocessResult, context: Context) -> None:
         """
          Handles warnings by extracting log issues, creating additional context, and calling the
@@ -491,6 +476,13 @@ def _handle_warnings(self, result: FullOutputSubprocessResult, context: Context)
         if self.on_warning_callback:
             self.on_warning_callback(warning_context)
 
+    def execute(self, context: Context) -> None:
+        result = self.build_and_run_cmd(context=context)
+        if self.on_warning_callback and "WARN" in result.output:
+            warnings = parse_output(result, "WARN")
+            if warnings > 0:
+                self._handle_warnings(result, context)
+
 
 class DbtRunOperationLocalOperator(DbtLocalBaseOperator):
     """
diff --git a/tests/operators/test_local.py b/tests/operators/test_local.py
index 94b0f8e27..728eea079 100644
--- a/tests/operators/test_local.py
+++ b/tests/operators/test_local.py
@@ -1,3 +1,6 @@
+import os
+import shutil
+import tempfile
 from pathlib import Path
 from unittest.mock import MagicMock, patch
 
@@ -28,6 +31,9 @@
 
 
 DBT_PROJ_DIR = Path(__file__).parent.parent.parent / "dev/dags/dbt/jaffle_shop"
+MINI_DBT_PROJ_DIR = Path(__file__).parent.parent / "sample/mini"
+MINI_DBT_PROJ_DIR_FAILING_SCHEMA = MINI_DBT_PROJ_DIR / "schema_failing_test.yml"
+MINI_DBT_PROJ_PROFILE = MINI_DBT_PROJ_DIR / "profiles.yml"
 
 profile_config = ProfileConfig(
     profile_name="default",
@@ -44,6 +50,20 @@
     ),
 )
 
+mini_profile_config = ProfileConfig(profile_name="mini", target_name="dev", profiles_yml_filepath=MINI_DBT_PROJ_PROFILE)
+
+
+@pytest.fixture
+def failing_test_dbt_project(tmp_path):
+    tmp_dir = tempfile.TemporaryDirectory()
+    tmp_dir_path = Path(tmp_dir.name) / "mini"
+    shutil.copytree(MINI_DBT_PROJ_DIR, tmp_dir_path)
+    target_schema = tmp_dir_path / "models/schema.yml"
+    target_schema.exists() and os.remove(target_schema)
+    shutil.copy(MINI_DBT_PROJ_DIR_FAILING_SCHEMA, target_schema)
+    yield tmp_dir_path
+    tmp_dir.cleanup()
+
 
 def test_dbt_base_operator_add_global_flags() -> None:
     dbt_base_operator = DbtLocalBaseOperator(
@@ -167,6 +187,7 @@ def test_run_operator_dataset_inlets_and_outlets():
             task_id="run",
             dbt_cmd_flags=["--models", "stg_customers"],
             install_deps=True,
+            append_env=True,
         )
         test_operator = DbtTestLocalOperator(
             profile_config=real_profile_config,
@@ -174,6 +195,7 @@ def test_run_operator_dataset_inlets_and_outlets():
             task_id="test",
             dbt_cmd_flags=["--models", "stg_customers"],
             install_deps=True,
+            append_env=True,
         )
         run_operator
     run_test_dag(dag)
@@ -183,6 +205,52 @@ def test_run_operator_dataset_inlets_and_outlets():
     assert test_operator.outlets == []
 
 
+@pytest.mark.integration
+def test_run_test_operator_with_callback(failing_test_dbt_project):
+    on_warning_callback = MagicMock()
+
+    with DAG("test-id-2", start_date=datetime(2022, 1, 1)) as dag:
+        run_operator = DbtSeedLocalOperator(
+            profile_config=mini_profile_config,
+            project_dir=failing_test_dbt_project,
+            task_id="run",
+            append_env=True,
+        )
+        test_operator = DbtTestLocalOperator(
+            profile_config=mini_profile_config,
+            project_dir=failing_test_dbt_project,
+            task_id="test",
+            append_env=True,
+            on_warning_callback=on_warning_callback,
+        )
+        run_operator >> test_operator
+    run_test_dag(dag)
+    assert on_warning_callback.called
+
+
+@pytest.mark.integration
+def test_run_test_operator_without_callback():
+    on_warning_callback = MagicMock()
+
+    with DAG("test-id-3", start_date=datetime(2022, 1, 1)) as dag:
+        run_operator = DbtSeedLocalOperator(
+            profile_config=mini_profile_config,
+            project_dir=MINI_DBT_PROJ_DIR,
+            task_id="run",
+            append_env=True,
+        )
+        test_operator = DbtTestLocalOperator(
+            profile_config=mini_profile_config,
+            project_dir=MINI_DBT_PROJ_DIR,
+            task_id="test",
+            append_env=True,
+            on_warning_callback=on_warning_callback,
+        )
+        run_operator >> test_operator
+    run_test_dag(dag)
+    assert not on_warning_callback.called
+
+
 @pytest.mark.integration
 def test_run_operator_emits_events():
     class MockRun:
diff --git a/tests/sample/mini/dbt_project.yml b/tests/sample/mini/dbt_project.yml
new file mode 100644
index 000000000..eaa39d188
--- /dev/null
+++ b/tests/sample/mini/dbt_project.yml
@@ -0,0 +1,20 @@
+name: 'mini'
+
+config-version: 2
+version: '0.1'
+
+profile: 'mini'
+
+model-paths: ["models"]
+seed-paths: ["seeds"]
+test-paths: ["tests"]
+analysis-paths: ["analysis"]
+macro-paths: ["macros"]
+
+target-path: "target"
+clean-targets:
+    - "target"
+    - "dbt_modules"
+    - "logs"
+
+require-dbt-version: [">=1.0.0", "<2.0.0"]
diff --git a/tests/sample/mini/models/.gitkeep b/tests/sample/mini/models/.gitkeep
new file mode 100644
index 000000000..e69de29bb
diff --git a/tests/sample/mini/profiles.yml b/tests/sample/mini/profiles.yml
new file mode 100644
index 000000000..0c53a3894
--- /dev/null
+++ b/tests/sample/mini/profiles.yml
@@ -0,0 +1,12 @@
+mini:
+  target: dev
+  outputs:
+    dev:
+      type: postgres
+      host: "{{ env_var('POSTGRES_HOST') }}"
+      user: "{{ env_var('POSTGRES_USER') }}"
+      password: "{{ env_var('POSTGRES_PASSWORD') }}"
+      port: "{{ env_var('POSTGRES_PORT') | int }}"
+      dbname: "{{ env_var('POSTGRES_DB') }}"
+      schema: "{{ env_var('POSTGRES_SCHEMA') }}"
+      threads: 4
diff --git a/tests/sample/mini/schema_failing_test.yml b/tests/sample/mini/schema_failing_test.yml
new file mode 100644
index 000000000..d97d1733d
--- /dev/null
+++ b/tests/sample/mini/schema_failing_test.yml
@@ -0,0 +1,18 @@
+version: 2
+
+seeds:
+
+  - name: mini_orders
+    description: This table has basic information about orders, as well as some derived facts based on payments
+
+    columns:
+
+      - name: status
+        description: 'Order status'
+        tests:
+          - accepted_values:
+              # this will intentionally fail, since the seed has other values for this column
+              values: ['placed']
+              config:
+                severity: warn
+                warn_if: ">1"
diff --git a/tests/sample/mini/seeds/.gitkeep b/tests/sample/mini/seeds/.gitkeep
new file mode 100644
index 000000000..e69de29bb
diff --git a/tests/sample/mini/seeds/mini_orders.csv b/tests/sample/mini/seeds/mini_orders.csv
new file mode 100644
index 000000000..31ac98589
--- /dev/null
+++ b/tests/sample/mini/seeds/mini_orders.csv
@@ -0,0 +1,10 @@
+id,user_id,order_date,status
+1,1,2018-01-01,returned
+2,3,2018-01-02,completed
+3,22,2018-01-26,return_pending
+4,9,2018-03-17,shipped
+75,69,2018-03-18,completed
+76,25,2018-03-20,completed
+77,35,2018-03-21,shipped
+78,90,2018-03-23,shipped
+6,68,2018-03-26,placed
diff --git a/tests/sample/mini/snapshots/.gitkeep b/tests/sample/mini/snapshots/.gitkeep
new file mode 100644
index 000000000..e69de29bb