Rotors wear slowly, so the question of whether it is safe to keep driving on worn ones depends entirely on how worn they are. Some wear is normal; past certain limits it becomes dangerous. Here is how to judge worn rotors.
Normal wear is fine
A rotor that has thinned somewhat but is still above its minimum thickness, and has a smooth, even surface, is perfectly safe to drive on - rotors are designed to wear gradually over tens of thousands of miles; see how long rotors last. Light scoring and a small edge lip are normal.
Below minimum thickness: not safe
Once a rotor wears to or below its stamped minimum thickness, it can no longer safely handle braking heat - it overheats, warps, and can crack; see rotor minimum thickness. A rotor past minimum should not be driven on - it is a genuine safety limit, not a suggestion.
Deep scoring, cracks, or heavy rust
Rotors with deep grooves from metal-on-metal (worn-out pads), visible cracks, or heavy pitting from long storage are unsafe regardless of thickness - see rotor scoring and why brakes grind. Cracks especially can lead to sudden failure. A big raised lip signals heavy wear too.
What to do
Measure the rotor and inspect it. In spec and smooth: drive on. At or below minimum, cracked, or deeply scored: replace before driving far - it is a safety issue and often takes the pads with it. Replace rotors as a pair with quality pads and fresh hardware.