Remi.b's answer is spot on - predicting the effects of losing a given species is nigh on impossible. However, I do think that there are some general trends that can give us some intuition, and make the campaign to protect certain species more than just an ethical issue.
Biodiversity (i.e. having a wide range of different organisms (and genetically diverse organisms)) has a number of potential benefits (for references see the work of David Tilman, among others):
- Improved resilience to catastrophic events.
- Improved temporal stability of populations within an ecosystem (which kind of dovetails with (1))
- Improved ecosystem productivity (this has mostly been studied in terms of the amount of biomass grown by plants in an ecosystem, but may generalize to other kinds of ecosystem functioning)
Ecosystem services are effects that various ecosystems have on the world that makes it more hospitable to humans. While some may be, as Remi.b suggested, overhyped, some are very real, such as bioremediation (the removal of compounds toxic to humans from the air and water), the continued presence of pollinators and nutrient-rich soil in which to grow crops, and carbon sequestration. Given that the continuation of these ecosystem services is necessary for our long-term survival as a species, it is in our interest to ensure that ecosystems remain intact to provide them. Increasing resilience and temporal stability of ecosystems reduces the chance that we will lose ecosystems and the services they provide. Increasing the productivity of an ecosystem in many cases increases the extent to which it can perform that service (this is most obvious in the case of carbon sequestration). (NOTE: I'm phrasing this all in terms of human survival because I think that's the most clear cut argument. I think there are other arguments, but they're murkier)
Of course, this raises the question of how much of an impact the loss of a single species (such as a tiger) is likely to have on the benefits associated with biodiversity. There are a few hypotheses here, but in any specific case the impacts will be fairly dependent on the role of the extinct species in the ecosystem. The loss of some species (keystone species) will have a disproportionate impact on the rest of the environment. While it is hard to predict which species will turn out to be keystone species until it is too late, apex predators (like tigers) are more likely than average to be keystone species, or to have an important but not disproportionate role in the ecosystem. This makes sense, because such predators are fairly often the only species at their trophic level in an ecosystem, and having more trophic levels dramatically increases the biodiversity possible in an ecosystem.
Even if there isn't reason to think that a certain species is relatively important to an environment, there can sometimes be benefit in trying to save it via what's called "the umbrella effect." This boils down to the idea that saving one species generally involves saving the rest of the ecosystem that it lives in. So sometimes, from a funding perspective, it makes the most sense to choose the most charismatic species from a given ecosystem and attempt to get public investment in saving that species. An attempt to save this species can then result in preserving a great deal of biodiversity in the rest of the ecosystem.
So, ultimately, no, we cannot know how important a given organism is to an ecosystem or to human well-being. But we can make some fairly well-informed guesses.