Trump also attended a sombre Monday ceremony honouring America's fallen, in Baltimore.
"If someone is wearing a mask, they're not doing it to represent what political party they're in or what candidates they support", the GOP governor said, as he started to become overwhelmed by emotion.
The images frame the stark contrast between an incumbent who wants to move on from the crisis, and a Democrat highlighting his trust in science and criticising the president for a bungled, inadequate pandemic response.
All 50 USA states have now partially reopened after a two-month shutdown.
According to the Centers for Disease Control and Prevention, a cloth face covering should be worn when people are in public, "especially in situations where you may be near people".
What they're saying: "In the face of COVID-19 we're seeing without any doubt how important unions are to this country - fighting for their workers to have personal protective equipment, for paid leave, and for safer workplaces", Biden wrote in a statement reacting to the endorsement.
Mr Biden later changed the picture on his official Twitter profile to one of him wearing a mask.
Asked how he would answer those attacks, Biden said: "Watch me". "Presidents are supposed to lead, not engage in folly and be falsely masculine".
Not only do Hannity's pleas stand in contrast with Trump's attitude on masks, they also mark a pivot from the conservative host's earlier efforts to downplay the pandemic itself.
New data reported on by United States media shows that the deadly impacts of coronavirus have been disproportionately felt in areas of the country that voted Democratic in 2016, like the coasts and big cities.
The mask itself has emerged as a political flashpoint.
"I'm not telling anybody what to do".
"Biden can wear a mask, but he was standing outside with his wife - flawless conditions, ideal weather".
"I think the people who out there frolicking need to realize that when you do that and you see no negative effect in one week, please don't be overconfident because the effect of spreading is not going to be seen for two, three or maybe even more weeks", he said.
Since early April, the Centers for Disease Control and Prevention has recommended that all Americans wear some form of face mask in public places where social distancing is hard.