Oil companies in particular need to worry about sustainability because their industry is seen as the antithesis of sustainability. With people worrying about the long-term availability of oil and the impacts of using oil, oil companies must adapt.
The most important thing for oil companies to do is to promote and participate in efforts to reduce the use of petroleum. They should publicize those efforts as seen in this link. This may seem like an odd thing for an oil company to do, but by doing so it helps to ensure that its business will thrive in the long term. This is true for at least two reasons. First, decreased demand leads to slower depletion of petroleum reserves, allowing the oil companies to stay in business longer and to go longer without having to try to exploit new (and probably more expensive) reserves. Second, promoting these sorts of programs will improve the image of the oil companies in the eyes of the public and reduce the risk that governments will intervene in ways that hurt those companies.