Many of the things that you might carry in your backpack are subject to flight restrictions. Look through FAA's chart and the TSA website, but this summarizes the specific items that are relevant to me:

item checked carry-on notes
flammable items
matches X Safety matches only (strike-anywhere are prohibited).
Maximum one book/packet.
Zippo/butane lighter X Maximum one lighter.
Micro-torches are not allowed.
Two additional lighters can be checked if placed in a DOT-approved container.
Zippos that are entirely empty (fumes purged too) can be checked.

Buy spares after landing.
lighter fluid X X (pdf)
fire-starting gel X X Buy after landing.
It's not clear if other tinder / fire aids are prohibited too. Probably better to buy these after landing.
camp stove ✓? Must be very thoroughly cleaned and have no traces of fuel odor. (html)
I always carry it on, because the TSA screeners don't necessarily know individual airlines' extra restrictions.
Some airlines have more restrictions: Some organizations recommend you always ship it, to avoid possible problems.
camp stove fuel X X (html)
Buy after landing.
sharp items
anything sharp —
pocket knives, hatchets, ice axes,
saws, other tools
X Must be sheathed or securely wrapped.
crampons ? The TSA says crampons can be put in carry-on, but chances are high that the screeners will disagree, so better to avoid problems.
trekking poles X (html)
other
spare/loose lithium batteries X All other kinds of batteries are fine in both.
(html) (pdf)



Items that are allowed / unrestricted: