I don't know of anyone doing it intentionally, but many people do leave the weeds in a pot and use their condition as one of many indicators of soil conditions.
To do it intentionally you'd have to know the needs of your sentinel plant relative to those of your tree, while remembering that it's taking up space and resources in the pot as well. Really there so many factors that would have to be taken into account that I don't feel it'd be accurate, or generally worthwhile.
Gardeners, and to an extent some farmers, do use sentinel plants in a was when growing in the ground, but this is generally just knowing the needs of the various things they have planted, and what deficiencies affect which of them first. They see something in one type of plant early on, and treat the whole plot to maintain a healthy balance.