Black Rhino, status: critically endangered
With around 5000+ black rhinos left in existence, and population numbers seemingly recovering (up from ~2400 in 1995), the black rhino is not really under any immediate threat.* The major threats to black rhinos is poaching (for the rhino horn trade), and, with prices for rhino horn increasing, it looks like it will be a problem for a while to come. Furthermore, corruption, war/civil unrest, easy access to guns, poverty, funding cuts (etc.) contribute to make the battle against poaching more difficult.
African Elephants, status: vulnerable
There are around 470,000 African Elephants left in the wild, and 25-30% of these are the Forest Elephant subspecies, so they aren't going anywhere soon.* Numbers are steadily increasing (~4% per year), although some populations are reducing. Poaching for ivory and bushmeat are signficant problems, but habitat loss and fragmentation is a massive issue. Elephants occupy a large number of habitat types and populations cover vast ranges, it is very difficult to protect an entire population range such that elephants could freely and safely move throughout their range (particularly at national borders), but things are getting better in many cases and efforts are ongoing.
An increasing number of transboundary elephant populations are co-managed through the collaboration of relevant neighbouring Range States. Large-scale conservation interventions are also planned through the development of conservation and management strategies at the national and regional level.
Polar Bear, status: vulnerable
Population estimates range between 20 and 25 thousand polar bears in the wild, and populations are on the whole fairly stable at the moment (in population studies, for populations where enough data was available, 1 grew, 6 were stable, 3 declined over a 12 year period). These animals are dependent on sea ice which, with human caused climate change, is rapidly declining. Climate change poses a more serious threat to polar bears. Relative to many things, like poaching and habitat fragmentation, it will take a very long time to reverse the tide; it may even be too late to prevent fatal damage to the polar bear habitat. They are also at risk from hunting, pollution, and habitat disturbance. Realistically, it is hard to predict when (or if) polar bears will go extinct because there is so much variance in the prediction of climate change and loss of habitat (sea ice).
Giant Panda, status: endangered
Less than 2000 pandas are left in the wild. Hunting and habitat loss remain to be enormous problems for wild pandas; masses of deforestation and fragmentation is occurring in their range. The chinese government has established >50 panda reserves, but this covers around ~60% of the population. The population trend is a decreasing one, which is serious cause for concern, and fragmentation of populations could reduce genetic diversity (increasing inbreeding depression and reducing adaptive capacity). Despite popular belief, pandas reproduce fairly well in the wild, it is in captivity that we have struggled, and that causes problems for breeding programs that can boost numbers and restore genetic diversity to sub-populations.
Mountain Gorilla, status: endangered
Fewer than 1000 mountain gorillas are left in the wild and populations are declining. Habitat encroachment, deforestation for the charcoal industry, bushmeat, and poaching are major issues affecting populations, and much has been fuelled by conflict (e.g. the 1990's war in Rwanda) in their range (in the last 20 years, 140 Virunga rangers have been killed trying to protect Gorillas). Also affecting them is human disease including the common cold which is often fatal to gorillas.
Bengal Tiger, status: endangered
Currently with a population of >2500, and with populations stabilising, and even rebounding in many areas (I was in Ranthambore last year, numbers have risen from ~20 to ~60 in the last 10 years), it seems the Bengal tiger is no immediate threat of extinction.* The major threats are poaching, population fragmentation, and habitat encroachment which leads to conflict with (and killing by) humans in the areas. Project Tiger is a major programme in India which has greatly contributed to successful growth in tiger numbers. The range of tigers is under increasing pressure from encroachment.
In summary, few models will be able to predict when or if any of these species will go extinct. Even if the short-term problems can be overcome, loss of genetic diversity prevents issues for the long-term prosperity of these species. Many of the problems are fixable in a relatively short time but will take a lot of effort and money, (such as poaching, hunting, deforestation) because they are mainly about cultural and social issues.
Climate change is perhaps more concerning because reversing the effects of human caused climate change will be extremely difficult (social and economic pressure), and slow to reverse; by the time we reverse the trend of rapidly declining sea ice it may be too late. Therefore, out of your species I'd say it's a close call between polar bears (despite good numbers they perhaps have the worst prospects of rescue) and mountain gorrilas (because numbers are so low and conservation projects are fighting difficult problems such as poverty and the effects of decades of civil wars and unrest). This article from national geographic suggests mountain gorillas may have as little as ten years left, while polar bears may be able to cope with some amount of warming, but the rate of warming is a major concern.** The WWF has also warned that pandas are not far from extinction.
What happens if we lose these species?
Many of these species are very attractive to the general public. The loss of some species would likely be worse than others; many of these are used to promote ecotourism and this would have varying effects for each. For the cases of black rhinos and african elephants many of these areas have many other species which are of interest to ecotourists (lions, leopards, cheetahs, white rhino, giraffe, crocodiles, buffalo, and many other mammals and birds, even sharks). However, if black rhinos went extinct then there would probably be losses in those species too (they face many of the same problems). Ecotoruism in the relevant areas tigers and pandas on the other hand are hugely important draws, India and China simply don't have the same diversity and abundance of highly sought after species with which to boost ecotourism should either go extinct (there are other nice species in these areas, but few are as highly sought as tigers and pandas). It's much the same for mountain gorillas and polar bears, these areas have less to fall back on should their species go extinct, so in many cases these species are of major economic importance.
That said, losing one or two of these species could cause some good. The outrage at losing some of the most popular animals in the world would be enormous. Funding for conservation and social pressure would likely swing massively more in favour of conservation efforts, making the fight to protect other species easier. Personally I don't hope we will ever have to rely on this strategy.
* Relatively speaking
** Strong selection may not get adaptive responses to occur quickly enough because generation times, standing genetic variation etc. such that while populations may adapt they don't adapt quickly enough.