I don't think that planting anything on top of the lilies is a good idea, but planting behind and between 2.5-3' tall shrubs and perennials will work. The taller varieties you grow, the taller hiding plant should be.
Simply cut your lilies after bloom to the top of the 'hider' and remaining foliage will work for recharging being practically 'invisible'.
Few examples,

Tall OT 'Touching' between 4' tall PJM rhododendrons
{{gwi:1036105}}

3.5' tall oriental 'Tarragona' behind 2.5' tall Ilex hedge
{{gwi:1036106}}

Very tall (probably about 7') oriental between shrubs

hi oyster2

You keep the stem after blooming so the leaves can strengthen the bulb by photosynthesis. Therefore you don't want to bury them in too much shade. So, keep as much stem as possible as long as they are green, but you don't have to keep them until they are all dried and shrivelled up. Once the leaves start to turn yellow and wilt they aren't contributing much more to the bulb and can be cut back to the ground without harm.

Ferns are very pretty interplanted with lilies and will stay green after the lily foliage turns yellow (some varieties are even evergreen).

-Helen