Is carboxytherapy safe for my dark circles?

At first, carbon dioxide should not be confounded with carbon monoxide, which is indeed a really toxic gas. A little more oxygen in the chemical structure makes the difference.

Carbon dioxide, the gaseous medium used in Carboxytherapy is safe, well known and has been used in medical procedures frequently and for different reasons. Large amounts are e.g. used to inflate the abdomen in laparoscopy for better view and access. It is produced by the body in the process of degrading carbohydrates and fatty acids to gain energy. So, no allergic or toxic reaction can occur.

During Carboxytherapy CO2 is injected by fine needles in different layers of the skin and subcutaneous tissue. In skilled hands, following all rules of antisepsis and after purge all remaining air from the tubing, some minor bruising may be the only side effect, but is rarely seen in eye treatments. 

Fast and stable results are seen by increasing oxygen level in tissue congested by "venous pooling", but causes for dark circles my vary.

Thank you for asking this important question.  Carboxytherapy is indeed a safe treatment and particularly effective at improving the appearance of dark circles around the eyes.  I have been providing Carboxytherapy in my clinic in Central London for a number of years and before that in my home country of Brazil, where Carboxytherapy is a very popular treatment and offered by many Dermatologists.  
I consider it a safe treatment in the hands of an experienced professional and so like any treatment, carry out your research to find a clinic that your are comfortable with before proceeding.  Best of luck with your treatment.