Hi. I have a real problem in removing tiger lilies, which are invasive even under even under patio stones.
The Hemerocallis fulva, also known as Orange Daylily, Tawny Daylily, Tiger Daylily – and tellingly, Railroad, Roadside, Outhouse, or Ditch Lily – is certainly showy, and it’s easy to understand why 19th century settlers enjoyed its robust growth and gorgeous swathes of bright colour. However, these plants are extremely invasive and very difficult to eradicate once established. They spread most effectively by their rhizomes, or tuberous root system, but can also self-seed, so it is helpful to remove all seed heads when they form.
Digging up the daylily rhizomes does work, but unless you are extremely careful in combing or raking the soil, this method usually does not remove the entire plant permanently, as daylilies can grow back from even small sections of root left in the soil. (Be careful where you dispose of the rhizomes you have dug up!)
Many sources say that the most effective way of removing these plants is to cut them right down so as little foliage remains as possible, and then smother them under plastic or a thick layer of mulch (several inches) which will prevent the plants from receiving either oxygen or light. Like any other perennial weed, they will respond by attempting to send up new growth, but with patience eventually the roots will die.
Since you are trying to eradicate plants that are pushing up patio stones, it may be that the best method for you would be to install a layer or two of thick dark plastic under your patio stones. Lift the plastic in the spring and assess the area; repeat if necessary.