I like Hart CO's answer about rounding and tax implications. But I'm gonna come at this from another direction.
Even one dollar per year might not be legal
A case can be made to challenge the legality of the one-dollar-per-year salary. The US Department of Labor (DOL) sets a minimum salary for exempt employees. Beginning on 01 January 2020, this minimum is $23,660 annually. In this context, "exempt" primarily means exempt from receiving overtime pay.
So if the employee in question is exempt, they would not be able to receive a salary of one dollar per year without violating that DOL rule. On the other hand, if they are non-exempt, they would be subject to the more well-known minimum wage -- currently $7.25 per hour in the US, and higher in some jurisdictions. Plus, they'd be subject to hour tracking and overtime. (I suspect most of these are senior executives, and they probably work well over 40 hours per week.) Most wage and hour regulations round to the nearest 15-minute increment, and have a minimum number of paid hours per shift. Even if the minimum is fifteen minutes, that's $1.81 even if the employee in question only worked 15 minutes in the entire year.
Per the Wikipedia article linked in the question, many recent or current situations of this phenomenon are government employees. Perhaps the DOL regulations don't apply to government employees -- I haven't found anything on that yet.
The special case of the U.S. President
In the specific case of the president, the U.S. Constitution sets forth that the president is to receive a fixed salary, and then 3 U.S. Code § 102 currently has it set at $400,000 per year. That's why President Trump couldn't opt for zero (or one dollar per year) and instead is donating his salary to various organizations.