This is unverifiable, but likely false. There are ~800,000 deaths/year from suicide, and there have been ~4 million deaths so far from COVID-19.
This is a common claim that has circulated numerous times during the pandemic. It is often used to downplay the seriousness of the COVID-19 pandemic. While sometimes it is used to raise awareness of mental health problems, it can also be used to imply that the ensuing mental health problems caused by the pandemic -- the effects of trauma, the loss of social contacts during lockdowns, and the stress of a worldwide crisis -- are more lethal than the actual disease.
This time the claim takes the form of a tweet from 2020, saying that there were more deaths in August from suicide than COVID-19. The location is not mentioned. The true figure is unverifiable for several reasons. Suicides are not always registered in official counts as "suicides" until the authorities have investigated. This can take several months, which causes a delay in official statistics. For similar reasons, there is a slight delay in registering COVID-19 deaths per day.
As the claim is currently circulating in the U.K., we will evaluate the claim for August 2020 and August 2021 in the UK, as well as mention the global figure. The best data suggests at the time that an average of 100 people a day were dying of COVID-19 in the U.K. August of 2021, and about 15 per day of suicide. This is an estimation, and that number may change as more suicides are registered after inquests conclude.
In August of 2020, there were 530 deaths where COVID-19 was on the death certificate. The Office of National Statistics (ONS) does not provide specific monthly data, but an annual figure of 5,224 deaths by suicide in 2020. An average gives us a figure of 435 per month.
August was a period in between waves in the U.K., and had a fairly low death total. The ONS reports that the total deaths from COVID-19 in 2020 in the U.K. was 72,178. This is more than fifteen times the total suicides in 2020.
This comparison between suicide and COVID-19 as causes of death does not reveal anything particularly useful. Suicide is not an infectious disease, and you cannot protect yourself from suicide with vaccines or masks. Similarly, coronaviruses are unlikely to be affected by their hosts' mental health problems. While the stress of the pandemic, and government responses to the pandemic, has certainly created or exacerbated mental health problems globally, it would be very difficult to establish any broad causal relationships on how exactly the pandemic has affected the number of suicides.
The COVID-19 pandemic has given rise to a lot of potentially dangerous misinformation. For reliable advice on COVID-19, including symptoms, prevention, and available treatment, please refer to the World Health Organization or your national healthcare authority.