I've heard that you can plant fruit to grow new trees, but I don't know how. Also, I've gotten a request to bury a time capsule for a resident, but I can't figure that out either. I've tried digging a hole with my shovel and dropping the item, but it just pops out away from the hole. How do I put stuff in the ground?
Dig the hole, then face the hole with your character and select the item in the inventory. Do NOT drag the item to the ground below your character on the screen; use the pop up menu instead. "Bury" will be an option if you're close enough to the hole.
Note you need to still be holding your shovel to bury items, even though you can fill in holes without the shovel.
If you still don't see the option, SteveKB may be right that you can't bury any more items; however if you've never buried anything in town before it's doubtful that you've reached this limit already, you can plant/bury many many things (I'm unsure what the limit is/if there is a limit in NL).
if you have too many trees or spots with buried items the game will not let you bury anymore. you have to cut a tree down or dig something up if you hit this limit and then when you attempt to bury something and click the item next to the hole you will see the option to bury it.
so yeah your town has just most likely reached the buried items/ trees limit I think.