Are belts necessary?

No, not really.
However, any martial art requires:

1. A syllabus
   - defining the components that will enable a student to achieve the objective
   - a path leading to the knowledge
   - a scheme of work
   - how the student will proceed
    
2. Topics
   - breakdown of the syllabus into modules of information
   - logical building blocks
   - small steps along the path
   - teach according to the students ability to learn
    
3. Discrete lessons
   - a lesson is an opportunity to explore a given skill
   - examples must be provided
   - thought-provoking
   - stimulating
   - encourage enthusiasm and participation
   - engage the student
   - allow for different ability levels
    
4. Proof
   - examinations, tests, grading
   - pressure-testing
   - an increasing scale of hardship

Without these, the material might potentially be taught in a haphazard, piecemeal, random manner.
Japanese martial arts schools are very good at offering a well-established means of teaching the art.
Chinese martial arts vary from school to school.
Tai chi schools in particular seldom have any kind of syllabus to speak of.

Are belts necessary?
No, but they chart progress.
The use of belts is a tried and tested method of teaching a detailed, complex syllabus in a logical, thorough, systematic manner.