"You can't push a rope" "You can't push a rope" is a wisdom saying that some engineering teachers pass along to their students.  Since I'm not an engineer, I can only guess at what they mean, but it sounds to me like code for a mathematical principle (but from control theory? numerical analysis? ergodic theory? dynamical systems? statistics?)
1) I would appreciate having any mathematicians who work with engineers hazard a general framework for a rigorous formulation of what the engineers mean by this slogan.
2) I would like to know of any deep theorems that (of the no-go variety) that naturally fall under this rubric (whether engineers know about these theorems or not).
 A: See http://engsci.unavoidable.ca/civ102/CIV102-Notebook.pdf pages 26 and 27 for a mathematical discussion of why you can't push a rope (from a civil engineering course).
A: It's not as mystical as you seem to think.  Simply, for a rope to be an efficacious component of a system that is performing work, the rope must be in tension (pulled on), not pushed on.  There is a similar saying among engineers:  sh-- doesn't flow uphill.
A: The rigorous mathematical context is stability. A straight rope in either tension or compression is a valid solution of the underlying PDE, but in compression this solution is unstable, so it cannot be realized in practice.
A: I've been an engineer, and now I am an attorney. I heard an engineer say this recently, and I thought about from both prospectives. As an attorney, I must disagree because there is an assumption that is left out of the saying, if inserted changes the conclusion:
You can't push with the LENGTH of a rope, only its width; and, 
You can't pull with the width of a rope, only its LENGTH. 
Think about placing a coil of rope between a stuck vehicle and another vehicle, as a cushion between them.
