According to current theory, the universe will eventually die due to what is known as the "heat death of the universe". This is not that universe will die because heat will get too large, but that the universe will die due to a lack of heat, i.e. the death of heat itself. When there is no discernible heat differentials anywhere in the universe, that means that there is no possibility of events happening, as every point in the universe will be at exactly the same heat level as every other point in the universe, i.e. absolute zero. Events occur as a result of energy (i.e. heat) moving from across places in the universe, if there is no heat, then there are no events, then there is no time either.
That's the classical view of heat death. However, if you take quantum mechanics into account, it's theoretically impossible for any part of the universe to be at exactly the same temperature, there will always be temperature fluctuations. So even if the universe approaches heat death, it will always be reawakened by another quantum fluctuation that will restart the universe. So in that case, time won't die, but time will get restarted eventually. So everything that happened before will get erased and restarted. This is time going on, but it's not the same time as we're in right now.