Does Lincoln County, Oregon require only white people to wear masks?

Lincoln county has published this statement:

In this Directive there were several exceptions identified recognizing that not everyone could or should wear a face covering: ...

People of color who have heightened concerns about racial profiling and harassment due to wearing face coverings in public.

So yes, they most certainly did.

No, there is currently no such requirement or exception to that requirement. For one week the health guidelines related to masks in Lincoln County, OR included an exception for people of color who were concerned about racial profiling/harassment for wearing a mask. That exception was removed in an update to the directive, and there are currently no exceptions related to race. Further, the Lincoln County Board of Commissioners issued a statement regarding this exception, and how they were revising the guidelines due the response they received.

However, in both versions of the health directive masks were not 'required' as the New York Post article claims, as the health directive explicitly states that law enforcement cannot do anything to enforce the directive.

Here are the health guidelines issued on June 16, 2020, which is the order referenced in the New York Post article, and the relevant text is at the end of page 2/start of page 3:

...all individuals in Lincoln County shall wear face coverings over their noses and mouths when they will be at (1) indoor public settings or (2) outdoor public locations and cannot maintain distrancing of approximately six feet from another individual who does not share their household.

...

The following individuals do not need to comply with this Directive:

• Persons with health/medical conditions that preclude or are exacerbated by wearing a face covering.
• Children under the age of 12. Children over the age of 2 but under the age of 12 are encouraged to wear face coverings but not required to do so.
• Persons with disabilities that prevents them from using the face coverings as described in this Directive. These persons must be reasonably accommodated to all them access to goods and services.
• People of color who have heightened concerns about racial profiling and harassment due to wearing face coverings in public.

This directive was updated on June 23, the day the New York Post article was published. The updated text can be found here along with some minor changes to recommended mask practices, the above exception for people of color was removed, and no comparable exception took its place.

Regardless, nobody is required to wear masks, as not wearing a mask is not grounds for law enforcement to get involved. The relevant text for this can be found on page 3 of both versions of the directive:

Violation of the Directive does not constitute or create grounds for private persons or public employees, including law enforcement, to stop, detain, issue a citation or undertake any other enforcement action under Lincoln County Code Chapter 10 against a person or persons who does not comply.