How do I compare a date field with the current time in a twig template?
If you feel like a preprocess function is overload and if you can afford to ignore timezones, try converting both date values to Unix timestamps to make them properly comparable. And prefer raw node values over formatted content values:
{% if ('now'|date('U') &gt; node.field_closing_date.0.value|date('U')) %}
  Past
{% endif %}