Wood has always been a limited resource, particularly in the colder climate of Europe. As time went on, the uses for wood got greater, whereas the forests got smaller.
Wood was the primary of providing heat and cooking over much of Europe. Vaclav Smil (‘Harvesting the Biomass’) gives statistics of typical fuelwood use in Sweden in the early 1800s as 5m3 per person per year, and 1.5m3 for warmer Piedmont in Italy. This is comparatively late, after various improvements such as slanted firesides and reflecting firebacks to improve heating efficiency. Smil has calculated that a pre-industrial city might need a woodland area more than 50 times the size of the urban area to provide fuelwood for that city – a major constraint on the size of the city.
Fuelwood was not the only use for wood. Many of the pre-industrial processes relied on the heat provided by kilns, fuelled by wood. Pottery was one, lime burning another. Another answer has already mentioned that in 17th century England, glass makers were forced to use coal instead of wood. The immediate reason for this was the demands of the Royal Navy. It took 6000 oak trees to build a 100-gun ship in the 19th century – and England had hundreds of ships (Smil again).
The iron industry was one of the biggest users of wood. This was compounded by the need to use charcoal rather than just wood, both the achieve the higher temperatures needed, and to supply the carbon needed for iron smelting. Charcoal is produced by heating wood to above 280°C.
In general, it takes about 5kg of wood to produce 1kg of charcoal, but the charcoal allows you to reach the sort of temperatures to smelt copper and iron. It takes about 90kg of wood to smelt 1kg of copper – this caused considerable deforestation across the Bronze Age Mediterranean. In 18th century England, 1kg of iron needed about 40kg of wood. A typical furnace would produce 300t of pig iron, needing a circle of forest of around 4km diameter. This was each year - a new area of wood would need cutting the next year. English iron production would have needed to consume about 1000km2 of forest every year. (Smil again.) England did not have enough wood – it was less than 15% wooded – and so imported iron from Sweden. England was short of wood: the price of firewood doubled between 1540 and 1570, and tripled again by the 1630s. (Priscilla Brewer: From Fireplace to Cookstove.)
What saved England’s economy was coal. The coal fields of North East England were both close to the surface and near to the sea. There was a flotilla of ships taking coal down to London from the 13th century onwards. The problem with coal is its weight: you really need ships to transport it, which limits where it can go. Coal was central to the Industrial Revolution in England, as it removed the energy constraints imposed by wood, but it needed a network of canals to be built to transport it, followed a few years later by the railways.
Even with coal, this did not immediately solve the iron-smelting problem, as burning coal with the iron produced too many impurities. The invention of coke by Abraham Darby finally allowed iron to be produced without charcoal.
England was the first European country to make extensive use of coal, because it has the biggest problem with wood supplies. By 1800, it was producing 80% of the world’s coal (Smil). Fossil fuels (coal) were providing more than half of England’s energy by the 17th century (Smil). France passed this mark in the 1870s, Russia in the 1930s.