I've been wondering about this for a while since on the 11th doctor's episode he mentioned something about jelly babies.
The Second Doctor started the trend.
Three liked them too, apparently.
It's not strictly canon, but the audio adventure Ghost in the Machine has Jo Grant find them in his pockets.
The Fourth Doctor used them as a running gag.
He's the most famous for them because the writers had him tossing the candies around all the time... sometimes literally throwing them.
This is the specific reference, based on the voice Smith is doing, in the episode where he channels the previous Doctors.
The Fifth Doctor...
...used them for barter in an audio adventure.
...had them in his pocket during The Mysterious Planet.
Seven liked them too.
In the TV Movie he ate them and carried them in his pocket.
Eight used them in the same film...
...he extolled their virtues as a method of distracting a police officer.