GCRL had the correct answers for two of the armbands.
The Yankees wore theirs to honor catcher and team captain Thurman Munson, who died in a plane crash on August 2, 1979. They wore the armband for the rest of the 1979 season and all of 1980.
The Cubs armband was worn in honor of team owner P.K. Wrigley, who died on April 12, 1977. That means that Donruss was using a really old picture. In fact, the set features 3 different Cubs players with pictures from 1977, Reuschel, Bruce Sutter, and Ray Burris.
The Braves proved to be the trickiest to figure. I found this list from the Hall of Fame, and it includes the Yankees and Cubs, but the Braves aren't listed.
At first I started looking around for any Braves player that died in 1980, the year before the Donruss cards came out. Then I realized that the uniform they are wearing was last worn in 1979 and I started looking around to see if any Braves players died during the season. Nope. Their owner was Ted Turner, and I knew he was alive, so I started digging around their front office.
Then, bingo, I found it. It was their general manager, Bill Lucas, who died on May 5, 1979. He was the first black GM in the majors and was only 43 when he died of a cerebral hemorrhage. The team wore the armband the rest of the season. The pictures that Donruss used had to be taken July 19-21, 1979, since the Braves other visit to Wrigley that season was in April, before Lucas died.