Steve Rogers is confirmed as being born in 1918 during the events of Captain America: The Winter Soldier.
When they enter the room where Arnim Zola's computerised brain is stored, Zola says something like:
Zola: Rogers, Steven: Born 1918.
Black Widow's year of birth (1984) is mentioned in the same scene as Cap's.
Zola: Romanov, Nataliya Alianova: Born 1984
(The quote might be a little off as I'm going from memory and a quick search couldn't get me the exact lines.)
In Infinity War, which is set in 2018, Bucky refers to himself as a "semi-stable hundred year old man" when Cap and the others arrive, indicating (not definitively) that he was also born in 1918 like Cap.
The memorial at the Howling Commandos museum exhibit lists Bucky as born in 1916 or 1917, depending on whether you look at the text or the bottom line (as noted by Aliden in a comment).
During the events of Spider-Man: Homecoming, Tony Stark refers to Peter Parker as being fourteen, to which Peter corrects that he's fifteen. If we know exactly which year this film occurred in (I'm not too sure given the “Eight years later” vs “Six years ago” issue) then we also know what age Spider-Man is.
A shot of the trailer for Spider-Man: Far From Home lists his birth date as being the 10th of August.
Nick Fury's date of birth is given on his ID card, shown in Captain Marvel (pinched from this answer) showing he was born on 4th July 1960.
- 1916/7 Bucky Barnes
- 1918 Steve Rogers
- 1960 Nick Fury
- 1970/1 Tony Stark
- 1980/1 Peter Quill
- 1984 Nataliya Romanov