You'd expect that to decrease positivity rates (by getting a wider pool of people to get tested), but instead the positivity rates --- the percentage of tests that are positive --- have been increasing.
But as I mentioned in another comment my pharmacist said she has barely sold any cough/cold syrup at all this year when by this time of the year they'd be flying off the shelves. The lockdown may have helped stopped common cold / flu spread as well. So my proposition may be insignificant.
My area is "minimal" in flu activity in the flu surveillance systems, but has increased COVID positivity rates from 3.3% to 5.4% in the past couple weeks, despite increased testing (and increased testing, in general, can be expected to push positivity rates down).
The people seeking tests were mainly two groups: people particularly concerned about COVID-- because of known exposure or COVID-specific symptoms, or people who occupationally are being tested because of surveillance.
If you water that down with people with the flu, you'd expect the positivity rate to go down: not that influenza carriers are even more likely to have COVID than those previous groups.
Influenza surveillance is showing "minimal" activity in my area, but a huge explosion in positivity rates for COVID-19 (3.3% to 5.4% in the past couple of weeks).