SPOILER ALERT: This post contains a minor spoiler for anyone who hasn't watched season 7 of Game of Thrones.
I'm going to go by the story in the show rather than the books for this post because, in the books, the Lannister gold mines have not been exhausted. In the show, they have no more gold and are forced to look for other ways of acquiring it. It's kind of a big deal because the currency of the Seven Kingdoms is very much on the gold standard.
Starved of gold to keep the economy moving, the Seven Kingdoms has already been doomed to a future of poverty.
There Is A Solution
Cersei collects taxes from everyone so that she can bling herself up and pay for other people to do stuff. Anyone who doesn't pay their taxes can have their head chopped off, which generally compels people to want to pay their taxes, even though they don't like her and her new haircut.
What if she collected taxes in something other than gold? What if she spent lengths of wood called tally sticks into the economy and told people that they could use them to pay their taxes?
So long as Cersei threatens to behead anyone who doesn't pay taxes using tally sticks, they're guaranteed to be used as money.
Addressing Conservation Concerns
Rather than chopping down entire trees, only a few branches would need to be cut off here and there. I haven't seen too many trees in the show, other than the ones along the Kings Road, so maybe there's scope to create forestry jobs for the unemployed. It certainly sounds better than having to resort to the Night's Watch.
Weirwood trees shouldn't be used as they're on the endangered list. However, Cersei may decide to target them as Bran Stark uses them to connect to the Internet. We need a Brotherhood With Green Banners to chain themselves to these trees and carry out newsworthy stunts that will capture the public imagination.
Conclusion
With control over her own sovereign currency, Cersei would never fail to make a payment, so there would never be any good excuse for a government shutdown (just like the USA, who also have their own sovereign currency).