Famously, they also burn.

Famously they also make houses and many other products like paper.

If you plant a forest where there was no forest before, then while individual trees will die, the entire forest will capture carbon dioxide proportional to its biomass.

Forests will typically grow for years-decades, and absorb carbon dioxide throughout, till their biomass (per km^2) reaches an equilibrium point. At that point the forest will go from a carbon sink to carbon neutral (averaged over several years to account for forest fires and subsequent regrowth).

Yes, they emit some carbon when they decay, but not all of vegetation goes back into CO2, so even mature forest is an overall carbon sink. New forest is an even larger carbon sink.

Estimates of the carbon sink per km2 are about ~200-400 Ton/km2/yr or ~2-4 Ton/ha/yr (Slope of curves in Fig 7 in https://www.nature.com/articles/s41598-021-99395-6) To absorb our current 10 GTon C (~30 GTon CO2) we would need ~30-50 M km2 of area; Arable (crop) land area is ~15 M km2 (~10% of Earth's total land area) and grassland is ~30 M km2. Clearly we aren't going to turn nearly all of arable/grass land area to forests, but vegetation management could be a significant part of carbon management.