No, mutually exclusive events are not always independent.

If events are mutually exclusive, their intersection is the empty set.

For example, if you roll a die, {x<4} = event that less than 4 turned up. If you look at two events, {x<4} and {x>3}, you'll notice that they are mutually exclusive. {x<4} = {1,2,3} and {x>3} = {4,5,6}. Their intersection is the empty set.

When two events are independent, knowing one does not affect the probability of the other.

Using the same example, the two events are not independent. If you know the roll is less than four, then you know that it can't be greater than three.