Feathers can be dyed just like hair or skin or leather or fabric; this is really common knowledge. Feather dyeing has been practiced all over the world for a very long time. Here is photo of a Hawaiian 'ahu'ula, or feather cape, made of dyed feathers and cotton for Queen Kapi'olani in 1882:
(Photograph by Wally Gobetz on Flickr; available under CC-BY-NC-ND-2.0.)
There is a beautiful pictorial in The Guardian showing a Spanish sub-culture of pigeon fanciers who race their brightly colored pigeons ("The Spaniards who paint their pigeons in Day-Glo colours – in pictures", with photographs by Rio Casas, 6 August 2014); this shows that not only feather take dyes, but that dyeing their feathers doesn't affect birds' flight.
(Cannot embed pictures here because there is no indication that sharing is allowed; you must go to the newspaper's website to see them; it is worth a click.)
The racing pigeon culture of Valencia and Murcia is discussed in "Flying high with Spain's neon-painted racing pigeons" by Sean O'Hanagan (same source, same date):
It is a mischievously drab opening shot for this surreal look at the racing pigeon culture of Valencia and Murcia in Spain, where breeders paint their birds in garish colours and release them to chase after a lone female. The pigeon that spends most time in her winged company wins.
There is such local prestige attached to owning a champion bird, hence the seriousness with which they pursue their hobby. From the ground, the men follow the flight of their individual pigeons – easily recognisable by colour – urging them on and rescuing them if, as often seems to be the case, they land on treetops or tall buildings or electric wires for a breather.
Since human-sized angels cannot possibly fly without mechanical help in Earth's atmosphere, it's moot to ask how much would their flight be hindered; it's pure fantasy anyway.