The smallest interval I can find is
2 seconds long:
9:59:59 - 10:00:01
My reasoning:
Whenever we change the seconds we also have to change the hours. That would result in a big interval.
So I thought it must have to do with some kind of overflow. I first thought about the overflow around midnight. 0:00:00 looked promising but the closest palindromic time before midnight would be 23:55:32.
Next thing I thought about was an overflow from ?:??:59 to ?:??:00. The only possible time ending in 59 is 9:59:59. Adding 1 second gives 10:00:00 which is no palindrome. Adding one more second gives the palindrome 10:00:01. Which is the smallest possible interval.