### Apache Airflow version

Other Airflow 2 version (please specify below)

### What happened

Greetings!
Recently I’ve faced a problem. It seems that assigning object, which can’t be serialized to JSON, to the dag_run.conf dict cause critical errors with UI.
After executing code example in "How to reproduce": 
Grid representation of the DAG breaks with following result:
<img width="800" alt="Pasted Graphic" src="https://github.com/apache/airflow/assets/79107237/872ccde5-500f-4484-a36c-dce6b7112286">
Browse -> DAG Runs also becomes unavailable.
<img width="800" alt="Pasted Graphic 1" src="https://github.com/apache/airflow/assets/79107237/b4e3df0c-5324-41dd-96f3-032e706ab7a9">
Dag itself continues to work correctly, this affects only UI graph and dagrun/list/
I suggest to use custom dictionary with restriction on setting non json values.

### What you think should happen instead

Raise an error

### How to reproduce

Execute following task. 
Composer 2 version 2.4.2
Airflow version 2.5.3
```
@task 
def test_task(**context):
    context['dag_run'].conf["test"] = np.int64(1234)
```

### Operating System

Ubuntu 20.04.6 LTS

### Versions of Apache Airflow Providers

_No response_

### Deployment

Google Cloud Composer

### Deployment details

_No response_

### Anything else

_No response_

### Are you willing to submit PR?

- [X] Yes I am willing to submit a PR!

### Code of Conduct

- [X] I agree to follow this project's [Code of Conduct](https://github.com/apache/airflow/blob/main/CODE_OF_CONDUCT.md)

dag = DAG("test_dagrun_no_deadlock", start_date=DEFAULT_DATE)
        with dag:
            EmptyOperator(task_id="dop", depends_on_past=True)
            EmptyOperator(task_id="tc", max_active_tis_per_dag=1)
        dag.clear()
        dr = dag.create_dagrun(
            run_id="test_dagrun_no_deadlock_1",
            state=DagRunState.RUNNING,
            execution_date=DEFAULT_DATE,
            data_interval=dag.timetable.infer_manual_data_interval(run_after=DEFAULT_DATE),
            start_date=DEFAULT_DATE,
        )
        next_date = DEFAULT_DATE + datetime.timedelta(days=1)
        dr2 = dag.create_dagrun(
            run_id="test_dagrun_no_deadlock_2",
            state=DagRunState.RUNNING,
            execution_date=next_date,
            data_interval=dag.timetable.infer_manual_data_interval(run_after=next_date),
            start_date=next_date,
        )
        ti1_op1 = dr.get_task_instance(task_id="dop")
        dr2.get_task_instance(task_id="dop")
        ti2_op1 = dr.get_task_instance(task_id="tc")
        dr.get_task_instance(task_id="tc")
        ti1_op1.set_state(state=TaskInstanceState.RUNNING, session=session)
        dr.update_state()
        dr2.update_state()