What would damage to the chordae tendinae in the left ventricle result in?

Mitral regurgitation. Damage to the chordae tendinae would result in problems with the closure and sealing of the mitral valve, and would lead to mitral regurgitation, or leaking, and possibly heart failure.
Chordae. The chordae are support structures for the mitral and tricuspid valves. When they are damaged, valvar insufficiency results. This can be severe enough to cause congestive heart failure either acute or chronic.