No, the word “tiger” should not be capitalised because it is a general reference to a large wild cat and a common noun. When you refer to a “tiger” you are referring to generic tigers and not specifying their exact species. But if you place before the word “tiger” the place where this cat is found you are specifying the species of tiger and you should capitalise the place name because place names are proper nouns (name for a particular person, place, or thing).
For example, if you want to be specific about a tiger you’d specify the species of tiger such as: Bengal tiger, Siberian tiger, Sumatran tiger and so on.
The same applies for all the wild cat species such as the African lion and Asiatic lion. Here the word “Asiatic” is capitalised too as it refers to a place, albeit a large area of the planet. Other wild cat species with places in their names are: European wildcat, North African wildcat (African-Asian wildcat), Asiatic golden cat, Chinese desert cat (note that this refers to the people of China but this is also capitalised), Canada lynx, Eurasian lynx, Iberian lynx, Andean mountain cat and African golden cat.
It is interesting to note that the great author and biologist, Dr Desmond Morris, who has written many books, sometimes capitalises the word “tiger”. Strictly speaking this is incorrect but grammar is quite flexible. It is not as if they are hard and fast rules which must never be violated on punishment of death!
At the end of the day it does not make a lot of difference because the important thing is to convey the meaning of what you want to say clearly and efficiently. People should not be hung up about grammar. The rules can be bent and they are elastic and changing. Grammar is just a convention, after all. They are artificial rules. If you look at ancient English it has changed vastly and in a thousand years the words on this page may look very strange.