Most block light sources work for any player carrying them. Strictly speaking they just need to be in the inventory to work. I will likely be adding a version soon that when loaded in hard mode will only work from the hotbar, but for now that's a little while away.
This is done by putting a light source block where the player is. which means its not as smooth as you might hope for though it does work fine in general. basically it's a form of dynamic lighting.
The light produced by the block when placed will be the same when held. Blocks like a furnace where they need to be lit do not produce light. the conduit and beacon also do not produce light when held. The only odd exceptions to this are the plain white candle which I let work unlit for the novelty. As well as amethyst shard which i elected to work as a light source instead of making the amethyst bud blocks work, which you normally don't carry.
Blocks that can't be placed underwater don't work underwater. So a torch will not light while underwater, nor will a lantern, but something like a glowstone or sea lantern will.
if you are holding multiple items that produce light. the best possible effect will be used. for example if you are holding a sea pickle, a torch and a lantern. the lantern will work as its the brightest untill you go underwater then the sea pickle will. BTW multiple sea pickles in inventory is still just count as 1. The same goes with white candles. colored candles do not produce light.