In most cases of MVP, exercising is encouraged.
However, there are big differences in severity of MVP.
Some patients have also regurgitation as a result of the prolapse. (leakage of the mitral valve).
Leakage is graded as none, trace, mild, moderate and severe.
When mitral regurgitation is more than mild, lifting heavy weights is not encouraged as it dramatically increases the pressures on the left ventricle (and on the mitral valve) during the maneuvers. Therefore, it makes sense not to pursue heavy resistance exercise as a precaution.
However, it is probably okay (not any real information in this area) to lift weights that are not so heavy (no strong bearing down and able to perform multiple repetitions) in order to enhance muscle tone and strength without building muscle mass dramatically. Low resistance aerobic exercises such as running, jogging, biking, etc are very good as well.
Imo (I am no doctor) push-ups and pull-ups/chin-ups fall in the multiple repetitions category and are no problem at all.