I presume you're taking the teleological view that symptoms are "good" and have a beneficial (if not fully understood) purpose, and therefore should not be messed with. That is a cognitive bias (a belief based on a construct, not objective evidence). Taking your example, I would ask you, what beneficial purpose does a runny/stuffy nose serve? Does it facilitate viral removal? Is it necessary to full recovery from a cold? Is long-term immunity to a rhinovirus enhanced if no symptomatic treatment is rendered?
Do you have any evidence that not treating symptoms is beneficial?
...why do we take medicines to suppress our symptoms?
First, let me clarify that symptoms are neither "good" nor "bad"; they merely inform (although one can argue that the absence of symptoms is a good thing.) Illness/disease/disorder is present. Thanks to them, now we know, and we can treat (or not treat) the underlying disorder. But the symptoms themselves are merely the body's reactions; they aren't necessarily good. Sometimes mild symptoms aren't treated, but the decision to treat is based on how disruptive (and/or dangerous) they are.
Let's assume only 'relatively benign' symptoms, for example a runny nose, sore throat and cough that typically results from infection with a rhinovirus (the most common cause of the common cold).
People treat a runny nose because it bothers them. It's hard to breathe easily with a runny nose, and the mere presence of nasal congestion causes a mild and very annoying feeling of air hunger in many people. Mouth-breathing dehydrates the oropharynx and upper airway, making a sore throat worse, worsens coughs, and makes sleep difficult, resulting in daytime drowsiness, irritability, etc. Overall, it's uncomfortable. The same applies to the sore throat: it makes eating, drinking, swallowing, and communication uncomfortable. Coughing (and sneezing) causes increased pain in sore throats, can keep people awake, spreads illness through aerosolized droplets (making for awkward social interactions), etc. The benefit of not treating these symptoms is largely unknown; the benefit of treating these symptoms (which supports the multi-billion dollar cold remedy industry) is that it just makes people feel better overall. They sleep better, cough less, feel less achey, swallow with less discomfort, etc. That's why people treat their cold symptoms.
The morbidity associated with non–influenza-related VRTI is not trivial. ...the total economic impact [in the US] of non–influenza-related [Viral respiratory tract infection] approaches $40 billion annually (direct costs, $17 billion per year; and indirect costs, $22.5 billion per year).
Some of the deleterious effects of rhinovirus infection are known (production of chemokines by epithelial cells resulting in influx of leukocytes into the airway leading to airway pathology; release of inflammatory cell products from neutrophils, cationic protein release from eosinophils, reactive oxygen species, etc., which can cause tissue damage.) However, the benefit of not treating a rhinovirus infection is unknown.
This means that unless and until benefits are shown of not treating symptoms, the risk to benefit ratio of treatment of the common cold is incompletely known. In that event, the practice will favor treatment.
My hope in answering this question is to shed light on the potential harm of teleological arguments, that they need to be discarded in favor of objective evidence. Things are not true because one believes them to be true; to quote Philip K. Dick, "Reality is that which, when you stop believing in it, doesn't go away."
This answer ignores obvious risks of treatment, such as decongestants in hypertensives, etc. which are present on information labels on OTC medications, and the risks of treatments without well-known benefit, e.g. Vitamin C supplements.
The common cold
The Economic Burden of Non–Influenza-Related Viral Respiratory Tract Infection in the United States
Role of Viral Infections, Atopy and Antiviral Immunity in the Etiology of Wheezing Exacerbations Among Children and Young Adults
How Viral Infections Cause Exacerbation of Airway Diseases