If it weren't for animals, insects, or people, would plants that produce seeds (nuts, fruit, etc) cease to exist?
No 

Not all seeds need to be ingested in order to be spread from one place to the other, in fact some won't grow at all if eaten, some need the fruit to help them germinate and sprout after they hit the ground. Nuts simply will not grow at all if you crack them open.

Most plants rely on the wind to disperse their pollen and seed.

Dandelions are a prime example of a plant with wind carried seeds, their seeds can travel for hundreds of kilometres on parachutes before settling down and germinating in the ground, plenty of other weeds and plants go to seed the same way.



Cotton bearing trees produce their cotton so the wind can carry their seeds, like the Cottonwood and many species of Poplar tree. 



One of the fun ones is helicopter seeds. Maples, elders and ash trees are among the species that disperse their seeds this way. The seeds will literally helicopter down, and fly great distances on the wind if it's strong enough. Even as an adult it's still fun to pick these seeds up of the ground and drop them just to watch them fly.



Some trees don't even need to go to seed in order to spread and multiply. Poplars are notorious for sending out suckers underground. Their roots systems spread out from the tree near to the surface and sprout suckers that will pop up out of the ground an grow into another tree. Other bushes and plants do this as well, like raspberry plants.

Not even all fruit is intended to be eaten, The fruit of the chestnut tree is sour, and covered in sharp spines that hurt. We used to spin them around and throw them at each other as kids imaging we were medieval warriors wielding maces. The spikes can get rather long and sharp, I've seen them stick to kids that have been hit by them, and after plucking them off they develop a golfball-sized welt covered in lots of tiny puncture marks.

 

Worst case scenario for plants that spread their seeds via burrs or by being ingested a pooped out elsewhere if animals were to suddenly cease to exist is that they wouldn't be able to disperse their seeds as far. They'd simply drop their seeds beside them, but they would continue to exist.