Legally the answer is "nothing" in the National Parks of Canada. I think the US is similar.
Ethically the best touchstone is the Kantian ethic: What would be the result of everyone doing this? As part of that, examine the renewal time, and the numbers of the thing in question and the number of visitors.
E.g. Taking the pine cone unless the pine is uncommon seems to me to be fine. A cone on the ground has already dispersed it's seeds. A cone is a byproduct. I'm removing a small amount of carbon from the environment. If campfires are allowed in an area, then removing a cone is not unreasonable. I would apply this to common shells on the beach too. They will soon be ground to dust.
I have several fossils that have come from our provincial forests. Mountain slopes where the fossil is common, or on eroding river banks. Tehir lifespan is short, once exposed.
Taking seeds of wildflowers again is fine -- my rule of thumb is to collect seeds from one plant in a clump of 100 plants, and to do so only a hundred yards from any trail. Even that would depend on it being a moderately low use area.
Removal of live plants is rarely successful, unless you have done your homework, understand their environment, and already have a created one back at the lab. This may be appropriate if you are a serious botanist trying to establish a breeding population of a plant. Again -- way off the beaten track, and never more than 1% of the visible population from wehre you stand.
A twig, a pebble, I don't see the problem, but I don't see the point either.