Is it true that a heart catheterization procedure is dangerous?

Yes and no: The complication rate for serious adverse events with a routine heart cath is on the order of 1/1000 to 1/10000. While rare, when they do happen they are quite serious, so you should always make the decision for a heart cath in conjunction with an experienced cardiologist.

There is some risk: The risk of diagnostic catheterization for the average patient is 0.1% or one complication in 1000 procedures. If the illness you are looking for is dangerous enough, this level a risk is acceptable to determine whether a serious underlying illness is present.