Yes it is a finite resource, fossil fuels will eventually run out if they continue to be used. Take oil, for example: oil comes from compressed plants and animals sandwiched in between impervious rock layers, and takes great heat and pressure to form. The world is consuming huge volumes of it, as if there is no tomorrow, yet we have already passed "Peak Oil" and no big new oil-fields have been found. Existing oil fields will become totally exhausted and the world will regress back to the 19th century (no cars, aeroplanes, plastics etc). Take coal: coal comes from compressed plants and animals in pervious rock layers. Unlike oil, coal will not run out worldwide, but its use will be reduced because of concerns about pollution, greenhouse gases, carbon sequestration, and the environment. Take uranium: although technically uranium itself is not a "fossil fuel" because it is not created from fossils, it is yet another buried fuel that is not being formed now. (Indeed, because of radioactive decay into other elements there is actually an ever-decreasing amount of uranium in the world.) For the making nuclear of fuel, only isotopes of Uranium can be used [I think it is the isotope U235 being converted into U238, but correct me if I am wrong], and uranium enrichment plants and nuclear power plants are constructed using equipment and processes only made possible with high-energy-use construction using oil or coal for their power source (for example, factories, foundries for steel and aluminium, computers). What this means is that although Uranium is not itself a primary fossil fuel, it is a secondary or even tertiary fossil fuel because it is dependent on coal and oil to create the products and processes to enable it to be used at all. Uranium is useless without fossil fuels.
It is not just fossil fuel that is finite: all fuel is finite. Indeed, the Laws Entropy (that everything is deteriorating and slowing down to Zero Point Energy) prove that , not just fossil fuel, not just every resource, but every single thing is finite and will end.