Can I do an internship in France on a tourist visa?

You need a visa to complete an internship in France. The visa you need depends on the duration of the internship. In your case, you need a short term visa since your internship lasts less that 90 days:

Short-stay visa

This type of visa is generally issued for tourism, business trips or family visits.

It is also issued to persons entering to France to take part in short training programmes, internships, conferences and corporate meetings, or to engage in remunerated activities (whatever the form), not exceeding 90 days.

In addition you need to be enrolled in education and must draft a written agreement with the host company or institution:

The conditions

Your project must be part of a training course organised in your country of residence (school curriculum, university degree or qualification) or an EU cooperation programme or an intergovernmental scheme in the fields of education, training, youth or culture.

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Your supporting documentation

In addition to the supporting documentation related to your personal situation as indicated in the visa wizard, you must include the training agreement (pdf - in French only) concluded between yourself, the training establishment and the host company in France.

Finally, note that if the internship lasts longer than two months you are legally entitled to a compensation:

Tout stage d'une durée supérieure à 2 mois consécutifs ou, au cours d'une même année scolaire ou universitaire à 2 mois consécutifs ou non, donne lieu au versement d'une gratification. Le stagiaire en entreprise perçoit obligatoirement cette "indemnisation" à partir de la 309ème heure incluse.