Yes, tigers do kill bears but it will happen quite rarely although there is conflicting information on the frequency of bear/tiger encounters. I’m going to quote a section of the best book on wild cats1, which provides reliable information in answering the question in the title.
“Feeding ecology: tigers will eat almost anything they can catch, from frogs to elephant calves. The menu includes birds, fish, mice, locusts, porcupines, moose, and monkeys but these animals generally form an insignificant part of the tiger’s diet.
There are also records of tigers killing and eating other carnivores, such as bears, leopards, lynx, wolves, and foxes but this is not a common occurrence either.”
It is the Siberian tiger which is more likely to come into contact with bears to attack and kill them (I would argue). Apparently, between 1944 and 1959, 32 cases of Siberian tigers attacking brown and Asian black bears were recorded in the Russian Far East, the home of the Siberian tiger. Bengal tigers also come into contact with bears in places such as the Ranthambore Tiger Reserve in Rajasthan, India.
Brown bears live in the open more than Asian black bears and are not able to climb trees and therefore they are attacked more readily by Siberian tigers.
The Wikipedia authors also state that tigers can tackle bears larger than themselves. They do so through their ambushing technique; jumping on the bear from an elevated position. It is said that tigers mainly feed on the fat deposits of the bear on the back groin and hams.
Siberian or Amur tigers do prey on adult brown bears and young bears but the percentages are low: 1.4% of the Siberian tiger’s annual diet is of Ussuri brown bears while 0.7% of their diet is made up of the smaller Asian black bear.
It is said that bears sometimes follow tiger tracks so that they can scavenge from tiger kills and to prey on tigers themselves. Although I am sure bears normally avoid tigers. Bears sometimes steal tiger kills and therefore the tiger’s presence is not altogether a bad thing for the bear.
It appears that an interesting study took place in the Sikhote-Alin region of Russia’s Far East (a protected area) in which 44 direct confrontations between bears and tigers were recorded. During these confrontations bears were killed 22 times and tigers 12 times. It could be said, therefore, that tigers in general win a tiger versus bear fight.
A report dated 1973 records the killing of tigers (including adult males) by brown bears. In all cases the bears ate the tiger.
I cannot vouch for the information provided on Wikipedia but the authors do refer to references in all cases. It is quite clear that the answer to the question, “do tigers kill bears?” is an emphatic, Yes. And encounters are not that rare in the Far East of Russia, judging by the records, which is where you will find Siberian tigers. Siberian tigers are the largest subspecies of tiger and the largest cat in the world.
Note 1: Wild Cats of the World page 351. ISBN-13: 978-0-226-77999-7