Fish and whale oils

They have been commonly used by many cultures across history for many purposes. If your people live on an island, I would be inclined to think that they would have access to fish as a food source.

https://en.wikipedia.org/wiki/Oil_lamp

Sources of fuel for oil lamps ... Also widely used were animal fats
(butter, fish oil, shark liver, whale blubber, seals).

https://en.wikipedia.org/wiki/Whale_oil#Source_and_use

Whale oil was used as a cheap illuminant, though it gave off a strong
odor when burnt and was not very popular.

https://en.wikipedia.org/wiki/Kudlik

This characteristic type of oil lamp provided warmth and light in the
harsh Arctic environment where there was no wood and where the sparse
inhabitants relied almost entirely on seal oil or on whale blubber.

Fallen tree limbs, dry leaves, pine cones, twigs -- "woody debris" in general. If they aren't managing their forests, they're going to be full of debris which will just pile up and make the place susceptible to forest fire. So perhaps it's part of their cult devotion that they gather the dead material up in order to tend to the living trees.

You could take it one step further and decide that it's not the entire forest that's sacred, but only a subset of the trees: say, all of the oaks, or all of the trees over 100 years old, or something. A forest in which everything is allowed to grow without maintenance is going to be an unattractive, tangled, mess. But a well-tended, managed forest could be full of pleasant groves and charming walking paths. It takes a lot of work, though. Foresters would cut down junk trees, bushes, shrubs, and prune the good trees to help them grow healthier, yielding plenty of burnable material as waste.

On the other hand, you could just give them peat or coal or petroleum on the island.

Do they have grasslands?

Then animal dung is used in a number of locations around the world. According to wiki, the places it has been used are: Asia, Europe, and the Americas.

If you're on the island of Ireland, the answer is peat. Many parts of Britain used it as fuel too. Of course this requires the landscape to be substantially peat bog rather than all forest, but if it's a large enough island then this shouldn't be a problem.

Coal

Have them use coal. The Aztecs used it, the ancient Chinese used it, the Romans used it. They can find surface deposits or dig rudimentary bell pits. Bronze Age tech is more than enough to mine ore deposits: that’s where they get the ores they need to make bronze in the first place.

Who needs fuel?

Bronze-age peoples made mirrors, so the real question is whether or not there's a bit-o'-metal on the island or even a hugely oiled and polished piece of hardwood that could be used to reflect light. Circle a pot-sized stone with such reflectors and you have a (believably) hot stone and plenty of heat to cook. It's not beyond the imagination that they would discover the value of making concave mirrors to focus the reflected light.

You also have a reasonably good defensive weapon — people will think twice after charging toward a bunch of reflectors (they have the sun god on their side!).

Natural Petroleum Seeps?

There are plenty of natural seeps of petroleum, essentially places where crude oil actually rises to the surface and forms puddles. These could be useful for fuel.

What if there are no natural seeps?

If there are no seeps, then perhaps you can have your people naturally discover underground petroleum. Many settlers in America encountered crude oil while drilling for brine.

As for the drilling part, simple cable-tool drilling doesn't require especially advanced tools, but does require a skilled operator (perhaps the art of making holes and bringing fire-sludge to the earthen world could become an art.)

What if they don't drill for oil?

In that case, there is still the possibility of coal. Coal is readily combustible, and finding it underground may involve some troubles (but nothing a little slave labor can't solve!)

What if they can't find coal?

If your people can't find coal, either, there are still a host of things they can burn. Whale oil is one; before the settlers of America discovered crude oil, whales were hunted, so that their copious amounts of blubber could be used as heating oil.

What if they can't kill a whale?

In any case, if there are other animals (terrestrial or marine alike), there is sure to be some fats on their body.

What if they're vegetarian?

There are still a host of things they can burn. These involve driftwood, grass, peat, dried kelp, and dried dung. These all burn to a satisfactory degree, although the dung source, collection, and preparation may be of questionable sanitary standards.

What if they just don't like fire?

If your natives are scared of fire, then mirrors are the solution. Bronze-age people made mirrors, but if they can't make mirrors, a simple transparent bag (jellyfish?) filled with water can act as a makeshift lens and heat things up.

Of course, a large parabolic mirror has frightening burning abilities, and if you were to ask me, I would stick with the fire.

Dry Dung Fuel

Essentially you take poop, dry it and then burn it later. This is a cheap and efficient fuel for homes and cooking that has been used throughout history and is still used today.

It works with both farm animal poop or human poop, so your villagers do not need to change much in order to use it, the only processing is to dry the fuel which can be done with just the sun.

There is a known issue with air quality when it is used to much but anything burned will have air quality issues. Assuming it is the bronze age it should almost be a non-factor.

Not only dry dung, but any dry organics is OK. If people are on the island, dry seaweeds are the best. You have heaps of them on the beach, and you even needn't enter the sacred inner land to collect them.

And you can play with strange stuffs that can be contained by the weeds and can influence something ( be a medicine, slow poison, a light drug)

Bronze Age means long distance trade

Making bronze requires smelting ores of (at least) two metals: copper and tin. Usable copper ores are much less common than iron, but they aren't extremely rare. Tin, on the other hand, is rare. Not as rare as the precious metals, but not all that far from it. China and Southeast Asia had (and still have) the most abundant sources of tin; Europe very few; and most of the rest of the world, practically none. Furthermore, in Europe tin ores were mainly found as pegmatitic "hard rock mining" -- especially granite, one of the hardest stones. (Asia had another advantage here: the deposits in Southeast Asia are alluvial. That is, the tin ores have been released from hard rock by erosion, and ended up in riverbeds, concentrated by their high density.)

That meant that the Bronze Age coincided with the development of long distance trading networks and the large scale stable empires that could support them; and that tin mining coincided with the development of serious masonry engineering.

So, does your island contain tin deposits?

1. No, this island does not mine tin. Fuel is only one of their problems. Being on an island, they will especially want timber for boats. With a need to import expensive tin, a shortage of boats, as well as fuel restrictions, you will need some way to explain how this culture could survive as a Bronze Age civilisation. It's not impossible, but they will probably need some other valuable export to explain how they fit in the trading network. Without naval power to protect it, you will also need to explain why others can't just take control of this resource.




2. No, this island does not mine tin, and it also isn't an economic power. Sorry, with all its disadvantages, I don't believe it is likely for your island to achieve Bronze Age technology.




3. Yes, it has significant tin deposits. You don't really need to worry about fuel; your island is the industrial heartland of a wealthy empire. Their trading network can supply them with fuel, food, and any other commodities they require. They also are probably building with granite masonry, not mud. However if the tin reserves are depleted, they will quickly collapse. Retaining control over this valuable resource will probably require serious military strength.

Wait, what about the fuel?
OK, I rabbited on about other issues without answering your actual question -- although others have answered it pretty well. Different cultures at different times have used a range of fuels for cooking and industry, and any of them might suit. Coal is probably the best, but it really isn't that big an issue. The one thing for which wood is absolutely irreplaceable for a Bronze Age civilisation is not fuel, but boats. Sure, you can make kayaks and coracles from animal skin, but you aren't going to haul tons of tin from Cornwall to Cyprus in a coracle.

With no timber, the only fishing will be casting nets from the shore, which for an island people means a serious reduction in resources, especially protein supply. There will be no trade, no cultural exchange. There will be no navy.

You also need to consider the impact of no land clearing. Sure, excessive land clearing will damage their environment; but having none at all will make it hard to grow crops, or pasture for livestock. Minimal fishing, no livestock, no trade, subsistence cropping between trees; your islanders will be desperately poor, and a few bad years could exterminate them.

Lava

Maybe there is volcano on the islan and there is hot lava inside - then you have as much heat, as you need.

If there is small, but stable lava stream going to sea (as infinite sink - at least for few decades, but even milenia, long enough to grow some technology around), the better - you can cook in the watter near it, use boulders to place there and then move elsewhere to provide temporary heat source, you can even make small side streams by blocking the main one, (until it overflows the sides at weekened points and then again stop, as the blocking boulder are melt and washed away) which can double for metal melting with some preparation, having sized peaces of metal, you can heat them over lava and then hammered (blacksmiths) to any form needed.

Also having lava drips somewhere could make for stone rods, which (after some manufacturing - stone-age techniques) may be of different use - from harpoon spikes, to maybe axis for wheels ...

Even some simple (stationary) steam engines can be build and filled with watter regularry to do some work - or just cooking, desinfection ...

Maybe there would be even some sources of sulphur (for basic alchemy) and watter plants, which are not sancted, so after drying could be used as fuel source as well as source of "wooden coal", so even black powder is not out of reach ...

Pushing wet trees of water plants to the lava and fast removing them can even provides some stone pipes (with burned watter plant inside, but with good training and practice it could be possible make some - then improve it by using these watter plants and sand to make them somehow regular for various use :)

In addition to animal oils, fossil fuels are always an option. Easily tappable crude oil, coal deposits, etc.

What they need is energy/fuel for fire. So they can:

1. Bake/cook/roast food

2. Have light in the night

3. Have a source of heat (for cold nights?)

4. Work on metal/forging

That can be replaced with:

1. Mirrored and centred light for heating a pot.

2. Have oil lights or have light bug pets in a cage

3. Use better clothing in the night

4. Hard... more mirrors?, but since the form is changing its had to focus the light on the metal, at this point coal isn't bad

In the ends they could also harvest organic material an collect and compress the decay gases, to make a gas grill

;)

Geothermal energy ?
The romans used to heat their houses with hypocausts :
https://en.wikipedia.org/wiki/Hypocaust

Using extremely hot water from hot springs could be used instead of fire.

Then you could use alternative ressources for other uses, but you can save a lot on heating fuel ressources.