This is a classical pseudoscientific claim: It is supposed to help against everything, including AIDS, Diabetes, Bronchitis, Cancer and so on. See for example this table I found online:

First, there is no real mechanism, and it is rather implausible to have an universal cure-all drug for all these completely unrelated diseases. There seems to be some weak evidence for the dental hygiene claims, but as the first reference writes, study groups where too small, not effectively blinded, so there is no final clue if this is true:
Oil pulling’s positive effects are based entirely on anecdote and not
at all on clinical research – because there’s been very little. Pubmed
lists six clinical studies related to oil pulling, all performed in
India, and their quality and results seem all over the map. Some do
indicate mild improvement in gum health, comparable to mouthwash use.
But they’re not well blinded, use very small sample sizes and involve
oil pulling only as part of a proper oral hygiene regimen. So it’s
impossible to tell what’s causing said improvement.
What clearly does not happen is the detoxification, as this is a pseudoscientific myth itself. Our detoxification is done pretty efficiently by liver and kidney, so there is no need to remove whatever toxin by a treatment.
See references 1-3 (especially 1) for background on the "oil pulling" and reference 4 for the detoxification.
References:
- Oil Pulling: Miracle Treatment or Woo Mouthwash?
- Oil Pulling Your Leg
- Is Oil Pulling Effective for Curing Medical Ailments?
- Fashionably toxic