Yes, the fossil-fuel industry benefits from subsidies equivalent to trillions of dollars, each year. Yes, the IMF's figures are broadly consistent with other studies: in particular, the cost of environmental damage from greenhouse gases alone, is of the order of trillions of dollars per year. These are costs which production incurs, but which are not borne directly by producers, and thus not reflected in the price. Environmental economics - for example within the IMF study quoted - treats these indirect subsidies as subsidies.
The industry benefits in several different ways. Merely taking quotes out of context, as the question does, hides the specific nature of the claim. However, the full report from the International Monetary Fund is explicit about the range of subsidies it considers:
the definition of subsidy
Here's what that report says in its abstract (my emphasis):
it focuses on the broad notion of post-tax energy subsidies, which arise when
consumer prices are below supply costs plus a tax to reflect environmental damage and an additional tax applied to all consumption goods to raise government revenues.
It goes on further on pages 10-11 to define subsidy in more detail.
post-tax consumer subsidies represent the amount by which the cost borne by the consumer falls short of the total economic cost of consumption. This excess cost (or subsidy) is either covered by governments in the form of budgetary support or foregone revenues or passed to the society in the form of environmental damage.
Producer subsidies exist when producers receive either direct or indirect support that increases profitability above what it otherwise would be (that is, the support is not passed forward in the form of lower consumer prices). This support can take many forms, including receiving a price for the output above the supply cost, paying a price for inputs below supply costs, receiving preferential tax treatment, or receiving a direct transfer from the budget
This is consistent with how the term is generally used within the field of Environmental Economics.
See, for example: The Economic Cost of Global Fuel Subsidies, Davis (2014):
Fuel subsidies are different from subsidies in most other markets because of the substantial external costs of driving.
And Energy subsidies: How large are they and how can they be reformed?, Clements et al 2014:
The subsidy estimates capture both those that are explicitly recorded in the budget and those that are implicit and off budget ... Tax subsidies
arise when energy taxation, which includes a corrective component for externalities and a component for revenue consideration, is below the efficient level. The efficient taxation of energy requires corrective taxes to capture negative environmental and other externalities owing to energy use.
And Fossil fuel subsidies in the Pacific island context: Analysis of the case of Kiribati, Peltovuori (2017)
Subsidy in this report is defined according to the definition of World Trade Organisation's (WTO) Agreement on Countervailing Measures (ASCM) (Uruguay Round Agreements, 1994). This definition has been recommended to G-20 by the Global Subsidies Initiative (The Global Subsidies Initiative, 2010). The methodology for quantifying fuel subsidies in this study is the price gap approach. In the price gap approach, a cost recovery price for fuels is first estimated based on the world free on board (FOB) market prices. All additional monetary and non-monetary costs such as transport, storage, distribution costs, and corrective taxes to account for externalities are then added on top of the FOB price. This cost recovery price is then compared to the actual price. The gap between the two prices is the amount of subsidy per price unit.
These pollution costs are included within the broader definition of subsidy as used by the IMF, because these negative production externalities are costs incurred by production, that the producer itself does not directly bear. Thus, those costs are not reflected in prices, leading to market failure and an unfair competitive advantage, of the same kind and market impact as if they'd received a cash subsidy on their directly-incurred costs.
The "Economics Help" site explains the issue
The deadweight loss from coal pollution is the red striped triangle: this comes about because coal producers do not have to pay for their pollution, leading to a competitive advantage, and an artificially high quantity sold Q1 at price P1.
The International Energy Agency looked at 40 developing countries (rather than the IMF which looked at all countries), and found the consumer-side domestic price subsidies (which are only a fraction of total subsidies) to be USD 325bn in 2015. The year before, that figure had been USD 500 bn.
The OECD looked at 41 countries, (34 OECD developed countries, plus BRICS + 2) and looked at some of the producer-side subsidies, finding them to be in the range USD 160-240 billion per year, 2010-2014
Global carbon emissions from fossil fuel in 2007 were estimated by Lawrence-Berkeley Labs at 7988 MtC (megatonnes of Carbon) (=8365 minus 377 from cement).
Leading climate impacts economist Chris Hope estimates the social cost of CO2 at around USD106 per tonne CO2 (tCO2), and one tonne of carbon (tC) is equivalent to 3.67 tCO2. Together, these put the environmental impact of CO2 from fossil-fuel emissions at:
7988 million tC/y x 3.67 tCO2/tC x USD106/tCO2 = USD3tn/y.
The IMF's definition of subsidy was the broadest, including the pollution damage costs, as well as covering all countries, and so is larger than simply the sum of producer- and consumer- subsidies for subset of countries plus some of the environmental impacts.
So, in total:
USD 160-240 bn/y: some of the producer subsidies (OECD)
+ USD 325-500 bn/y: some of the consumer subsidies (IEA)
+ USD 3tn/y: some of the environmental impact (Chris Hope / LBL)
So the IMF estimate of USD 5tn/y (all countries, all producer and consumer subsidies plus all environmental impact) is consistent with this.
The size of the greenhouse-gas impacts alone is in the trillions. This means that the lobbyist's opinion-piece in Forbes is irrelevant to the claim in question: the aspects of the subsidy included, to which the lobbyist objects to, are tiny compared to the environmental impacts - which even they agree are valid.