My understanding is that the forests didn't give any financial advantage - they were essentially hunting preserves. It seems like cutting this down a bit would have been a fairly cheap way to satisfy the magnates of the realm. So why was he so reluctant about it?
Forest was actually very valuable, both financially and in terms of prestige. First it's worth noting that forest in the medieval sense doesn't necessarily mean a heavily wooded area. Forests are better thought of primarily as a game preserve, one which would have wooded and scrubland areas.
This is an extent (a record of property and its value) from 1338, only 21 years after Edward's reign:
"Item CClx. acre terre, pretium acre vj d. Summa vj li. x s.
Item XV. acre prati, pretium acre iij s. Summa xlv s.
Et xxiiij. acre pasture, pretium acre xij d. Summa xxiiij s.
Item xl. acre bosci, pretium acre xij d. . . . . xl s."
"Also 260 acres of meadow, each acre [is] 6 pence. Total £6 10 shillings.
Also 15 acres of meadow, each acre [is] 3 shillings. Total, 45 shillings.
And 24 acres of pasture, each acre [is] 12 pence. Total 24 shillings.
Also 40 acres of woods, each acre [is] 12 pence. [Total] 40 shillings."
Taken from Lambert B. Larking and John Mitchell Kemble (eds), The Knights Hospitallers in England (London, 1857), p. 81.
We can see from this that arable land was the least valuable, with pasture and woods equal, and meadow the most valuable. Land values could of course vary by region and quality. This particular account is for Norfolk. Forest in Hampshire, for instance, may have had a different value. The quality of the land is also not given, the meadow may be exceptionally good, whilst the normal arable land quite poor. However, this document shows that at least for some areas, woodland was the joint second most valuable type of land. It should also be noted that forests often included meadows and pasture in addition to woodland, making them potentially even more valuable than the figure given for woodland alone suggests.
In addition to this pure economic value, they also had much political and symbolic use. Hunting was a mark of social status and so the right to hunt in royal forests could be sold or traded by the king to his vassals. If someone was granted land or rights 'in libera foresta' (in free forest), they were given full rights over the 'vert and venison' of the forest, meaning all the plant and animal life in the area. Not only would the grantee control a supply of a timber and broom (used to thatch roofs) but also have rights over hunting the local deer and other animals.
You can see how highly-prized such rights and lands were by how the Crown and other landowners protected them. The woodland in forests was often encircled by earthworks and fences to protect the trees from livestock. In England and Scotland there were foresters tasked with maintaining the king's rights over their forests, and the rights of those the king favoured. They also managed the forest courts, which would prosecute people that had violated forest law (such as by hunting where they had no right to). Foresters were given some benefits for their service, liking being excused from rent or having the right to collect timber. These foresters were often very unpopular, they were regularly accused of corruption, and even when faced with evidence of the claimant's right, they could still be resistant. A court case in 1256 in Scotland saw the Abbey of Lindores take an officer of the earl of Strathearn to court. The monks had been denied access to the earl's forest of Glenlichorn, where they harvested timber. Despite having written evidence showing they had this right, the officer only grudingly assented, and required that he or one of his men be informed everytime they monks took any wood.
Sources:
Anne Crone and Fiona Watson, 'Sufficiency to Scarcity: Medieval Scotland, 500-1600, in T. C. Smout (ed.), People and Woods in Scotland (Edinburgh, 2003), pp. 60-81.
Andrew Pattison, Power Relations in the Royal Forests of England (Unpublished MA thesis, University of Oulu, 2013): http://jultika.oulu.fi/files/nbnfioulu-201312052004.pdf
Tom Williamson, Environment, Society and Landscape in Early Medieval England: Time and Topography (Woodbridge, 2013).
I had the impression that forests did give financial advantages, through fines levied against people who offended against forest law in a myriad of ways (and forest law was far more dependent on the king that normal law), and through fees for people who wanted more privileges, like assarts. Does anyone have evidence either way?