Has there ever been a situation when birds caused damage to a rocket? At what speed would birds be a problem?
As noted in Johnny Robinson's answer, there has been at least one recorded incidence of bird strike during launch. This was STS-114, the first Shuttle launch after the fateful STS-107 mission. That bird most likely was a turkey vulture, average mass about 2 kg. The piece of foam that hit the wing leading edge of the Columbia during the launch of STS-107 weighed less than half that.
Bird strike was a concern throughout the Shuttle program, given the large number of large birds (e.g., vultures, eagles, hawks, herons, egrets) that live in the vicinity of the Kennedy Space Center. A 1988 study by Karen Edelstein and Robert E. McCarty, "Space shuttle orbiter windshield bird impact analysis." ICAS, 16th Congress. 1988, found that
The forward window assembly for the orbiter offers insignificant resistance to a 4 lb bird impact.
The critical velocity for a 4 lb bird strike on the forward window was rather small, anything above about 200 miles per hour. While the study focused on landing rather than launch, bird strike during launch would also have been problematic had the bird strike occurred on a window. During launch, the Shuttle quickly got up to speeds that would have enabled a large bird to break through the forward window. A bird strike on a wing leading edge would also have been problematic.
NASA took the STS-114 incident to heart. It aggressively stepped up its abilities to detect birds in flight, started a launch day vulture "trap-and-release" policy, and developed policies to quickly remove roadkill that attracts vultures from the Kennedy Space Center.
The space shuttle crew closed and locked their visors during first stage to protect against bird strikes breaking the windows and depressurizing the cabin.
They opened them after SRB sep because at that altitude bird strikes were unlikely, and because the suits leaked O2 into the cabin when visors were down.
No bird ever broke the windows.
Broken windows could have caused the crew to abort depending on the type of damage and the mission elapsed time.
Apologies for the lack of references, I am away from my usual resources.