I apply a filter to a smart chute, save it in a schematic, and when I load that schematic in the filter is gone.

I assume this is because the smart chute can store an item inside of it, so if it had the #create:safe_nbt tag it would allow the player to duplicate items and transport items between worlds.

Can confirm.
Create v0.3.1a

Video below shows the funnel's filter correctly copying, but not the smart chutes filter. It DOES appear in the schematic though, and if printed through creative. Also not just visual.
https://user-images.githubusercontent.com/73862885/113668480-6846cb00-9706-11eb-8789-0c1cf962c128.mp4

It is NOT the smart_nbt tag that saves filters, that is used for stuff like signs where ALL the nbt is always gonna be ok to copy (probably to provide a way for other mods to add support)

Seems like someone just forgot to copy some code over to the smart chute or include the smart chute in a list of blocks who have a filter slot.

That makes sense. I was assuming there would be a way to copy part of the nbt, but not all of it. On a related note, aerial faith plates or whatever they're called in Create do not save launch data when printed with a schematic and always reset to shooting you to the block in front of them.