There is an interesting bit from "The Way to Eden". The TOS Enterprise's presence across the RNZ would be viewed as a "military intrusion" and not a "exploration intrusion" according to Kirk.
SULU: Captain, we're now leaving the Neutral Zone. Bearing into Romulan space!
KIRK: Any patrols?
KIRK: There will be soon. Doctor Sevrin. You are violating Romulan space and endangering the peace of the galaxy. The Romulans will view this as a military intrusion and attack. Bring the ship about now.
But we do know that some ships, particularly ones that move some illegal cargo like Romulan Ale go back and forth across the RNZ according to McCoy in "The Wrath of Khan".
McCOY: I only use it for medicinal purposes. I got aboard a ship that brings me in a case every now and then across the Neutral Zone. Now don't be a prig.