In this case, the word "is" marks a dividing line between the the subject and it's predicate. Once we have crossed this line it is too late to add additional supporting clauses to the subject. For example, the following would not work "Mary, my sister, is taking the dog for a walk, who likes carrots". To make this work, we must introduce the supporting clause "who likes carrots" before the predicate: "Mary, my sister, who likes carrots, is taking the dog for a walk".
The tricky thing with your sentence is that "is 70 years old" is not an obvious predicate. As a descriptor (as opposed to a definitive action like "is taking the dog for a walk") it may seem like just another supporting clause, but it's not. It is the predicate and supporting clauses introduced after the predicate are, in fact, too far from the subject. Your teacher is correct.
To fix this, we can either move the predicate to the end, after all of the subject's supporting clauses:
My neighbor Mr Lee, who plays basketball every day, is 70 years old.
Or make "is 70 years old" a supporting clause and "plays basketball everyday" the predicate:
My neighbor Mr Lee, who is 70 years old, plays basketball everyday.