The short answer is no. Even in areas where COVID-19 is spreading rapidly, there is no evidence that dogs and cats are suffering from COVID-19 at the moment. However, we are in a fast moving process and changes in this situation need to be monitored and checked.

A few days ago it was published that a dog in Hong Kong whose owner had been infected with SARS-CoV-2 was tested mildly positive for the virus in a swab of the mouth and nose cavity. The dog showed no symptoms of the disease. Since the test methods used are very sensitive and can detect very small amounts of the virus, the positive result could also be due to contamination of the dog's nasal and oral mucosa from the virus-containing environment.
The already known coronaviruses of dogs and cats (canines coronavirus and felines coronavirus) can be clearly differentiated from the currently circulating SARS-CoV-2 and do not pose a known risk to humans.

Is my pet a potential risk for SARS-CoV-2 transmission?
There is currently no evidence that pets play a significant role in COVID-19 transmission. However, general hygiene measures should always be followed when handling pets to reduce the risk of transmission of pathogens between pet and humans.

Further information: Information leaflet Animal Hospital, Tierspital, Vetsuisse-Fakultät Zürich