The redstone machine looks like this (you may also see the following image):
.L..R..O→O↓
.A..B..D..N.
A=air, B=any solid block, D=dispenser, L=redstone light, N=bee nest, O=observer, R=redstone dust
And the effect looks like another image.
After a bee put honey in a nest, causing a block state update, the machine starts squirting honey bottles and the nests' states switch between honey_level 0 and 5.
I also did another experiment and that indicates that when a nest(in any honey_level) is in front of a dispenser consists of empty bottles, the dispenser will "collect" honey from nest and also may shoot it out.
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How did you get the bee nests? Are there any bees inside of them?
Also, do any error messages appear in the game output log?

I got it from the inventory in creative mode. This happened not long after I placed them, most of them are empty...I suppose.
No error. The following warn has appeared 8366 times in 100 seconds.
[Server thread/WARN]: Keeping entity minecraft:bee that already exists with UUID 1f3eb096-dcbb-4aca-b133-4089f77abf79
Looks like you copied a bee hive with NBT. This is a known issue, see MC-159383.