Teachers didn't require any training or licensure at this time.
The U.S. Department of Education has published a brief paper on the history of teaching in the U.S. here. Since Tom Sawyer takes place in the 1840s, there is really no chance that a college education would have been required, except in some extraordinary local circumstances. Mr. Dobbins may have wanted to become a doctor, but couldn't afford the necessary and expensive medical training.
During the 19th century, no formal educational requirements were in place for teachers. Teacher certification was inconsistent, but there was no "teaching profession" and no formal education was required. You just needed to persuade your local government that you were good enough to be a teacher.
Teacher certification in the nineteenth century was irregular and diverse. There was no single pattern, and there was no teaching profession as such.
In the first half of the nineteenth century, the requirements for entry into teaching were modest: new teachers had to persuade a local school board of their moral character, and in some districts, pass a test of their general knowledge. In 1834, Pennsylvania became the first state to require future teachers to pass a test of reading, writing, and arithmetic. By 1867, most states required teachers to pass a locally administered test to get a state certificate, which usually included not only the basic skills, but also U.S. history, geography, spelling, and grammar.
At this point in history, some states had required basic skills tests. However, more than likely any requirements would be imposed by the local district and not by the state.
It wasn't until the 20th century that formal teacher's certifications appeared in the United States. During this time there was an increasing professionalization of teachers. Pedagogy degrees (like a bachelor's degree in education) appeared, and eventually became the norm for teacher certification.
Physician's training was expensive and needed in practice.
Physicians' training on the other hand, was both expensive and required in practice (though perhaps not formally required).
According to the U.S. National Library of Medicine, medical licensure in the U.S. dates back to the colonial period (that is, prior to the American Revolution). By the 1800s medical societies (essentially professional organizations) were regulating medical practice and certifying physicians. Your regional medical society would be responsible for certifying you as a doctor. Universities and medical societies both offered medical degrees.
Slawson(2012) summarizes medical training in the United States throughout this period. Medical degrees typically consisted of 2 years of lecture and 3 years of apprenticeship. This cost about between about $790 - $1,250 in total, at a time when the annual income for a factory laborer was about $360 a year - half that for rural labor. Unless your family already had money, it would be difficult to afford this education.
It seems that actually having a medical degree was often not necessary for practice. Perhaps only 40% of practitioners had medical degrees (though they may have attended lectures and had some apprenticeship before their own practice). Nonetheless, affording this education at all would have been difficult for a laborer's family.
