World Council of Churches, Unitarian Universalists, United Church of Christ, and many smaller/regional church denominations and affiliated organizations have established climate divestment policies. Others are percolating - the Methodists are studying their investment policy, the Presbyterians are first going to try to persuade the companies to give up their core business model (good luck with that!)* and then we'll see them and others consider this issue.
People involved in climate divestment and had also been involved in South Africa divestment a generation ago say that climate divestment is moving faster. An Oxford study backs that up (p. 11).
The same study acknowledges limited direct financial impacts of divestment except for coal industry, but then focuses on the stigma issue:
As with individuals, a stigma can produce negative consequences for an organisation. For example, firms heavily criticised in the media suffer from a bad image that scares away suppliers, subcontractors, potential employees, and customers. Governments and politicians prefer to engage with ‘clean’ firms to prevent adverse spill-overs that could taint their reputation or jeopardise their re-election. Shareholders can demand changes in management or the composition of the board of directors of stigmatised companies. Stigmatised firms may be barred from competing for public tenders, acquiring licences or property rights for business expansion, or be weakened in negotiations with suppliers. Negative consequences of stigma also include cancellation of multibillion-dollar contracts or mergers/ acquisitions. Stigma attached to merely one small area of a large company may threaten sales across the board.(p. 14, citations removed)
The stigmatization from divestment will have financial consequences. These companies will have to pay more for employees and for other businesses to work with them. Companies with a toe-hold in the fossil fuel sector will find it better for them to get out.
Most important is that stigmatized industries will find it tougher to manipulate the political sector. That's one reason why they disguise their funding, but the disguise is imperfect, and the difficulty gets worse with the stigma.
Two other points. The study acknowledges political restrictions resulting from the climate divestment effort could destroy the perceived value of reserves that end up staying in the ground. When the carbon bubble pops is hard to predict, but any downward pressure increases the possibility of it happening soon.
Second, when companies divested from South Africa they weren't required to physically blow up the businesses they left behind - they sold them. The argument that it had no financial impact was around then, but we see what happened in the end.
*I think there is a business case that fossil fuel companies should 1. stop wasting money exploring for new reserves, 2. sell the reserves they're not going to be allowed to develop before the carbon bubble bursts, 3. play out the remaining and cheapest reserves and 4. either distribute the profits and wind down their companies, or invest in another business model. Not likely to happen, though.
I'm ignoring the complications of when natural gas can substitute for coal.