Keep them growing for as long as you can while they are green. Once most of the leaves and stem start to turn yellow, it is ok to cut it back.

I agree with the first poster. Also feed the lily once the flowering is done. Foliage feed or give some bulb food after flowering. It may be only my soil but without the fertilizing I don't get as big of blooms the following year as I do when I fertilize. Early spring when they first poke up give another helping of the bulb food.
I leave a short section of the stem up so I can find the lily the next spring to apply the fertilizer.

I cut back the stem by 1/3 and let the remaining 2/3's die back. Sometimes I leave the dead stem in the ground all winter...it helps me remember to not plant there in the Spring!

Last year I tried an experiment and planted thunbergia nearby...the vine used the lily stalks as a trellis and climbed up to the top.

Ciao,
OITGAD