Lilium lancifolium is the one most commonly called tiger lily. The good news is that it is probably the most virus tolerant of any lily. The bad news is that 99.99% of all Lilium lanifolium have viruses. The good news is that it is very rare indeed, that any virus symptoms ever physically manifest on L. lancifolium. You never would know it had any viruses by looking at it. The bad new is... so what, the viruses can still transfer from your infected Lilium lancifolium to any other lily.
Everything I have said thus far, is not the result of your pictured symptoms; rather, it is true for all Lilium lancifolium.
For every plant, animal, fungi, etc., there are many different viruses that can infect, and this is no different for lilies. There is something we call a virus load, and this can be a high concentration of a virus's particles, or the presence a high number of different kinds of viruses. With lilies, it is usually the latter, and my best guess is that this is what is going on with your plants: they have so many different viruses attacking that it is finally showing a symptom, but the species is so resilient, that it still does not reduce the overall health.
whether I am right or not is debatable, but the action needed is not. You need to get rid of those plants because of what I stated in the first paragraph here, and they are planted so close to other lilies that are susceptible. Realize that all those little black bulbs that form at the junctures of leaves and stem are automatically infected, too. You will need to be very fastidious about the removal of all these little resulting plants, because they, too, will be sources of inoculum to infect your other lilies. This is doubly important since you say you have a good population of sucking insects (aphids) that transmit the diseases.
If you choose to poison the lilies, know that it will take multiple applications. The lilies will take in the poison and distribute it throughout the roots and large mass of the bulb. The concentration of the poison will become so dilute that it will not be strong enough to kill, hence multiple applications will be needed. Likely, you will spray, the poison will look like it is working, and then the plant starts to look better again. That will be your cue to spray again. Do not be tempted to spray at a higher dose than recommended, as this will likely burn the foliage and the poison won't be taken in at all.
