Can I live in Thailand for 2-3 years while studying online as a Danish student?

Not gonna work.

No

The short answer is no, you can't obtain an educational visa for the purposes of taking a Danish educational course even if it's online.

You can get an education visa (there're a number of requirements needed) and these do require you to physically attend an endorsed educational facility, the normal courses undertaken are obviously learning Thai language although you can also enrol with a Thai university for an undergraduate degree course for example.

Educational visas also require you to extend them every 90 days.

“With tourist visa, travellers are strictly prohibited to conduct in all kinds of work including volunteer work and internship.”

From Royal Thai Embassy Copenhagen, which should have been the thing to figure out, before dreaming tha almost impossibel.

2-3 years is highly likely unless you get a job with a Thai company, marry a Thai citizen, or have a half Thai child.

Your best bet is probably an education visa to study Thai language or Muay Thai. You can get 1 year, maybe 2 years maximum using this. You will be expected to attend classes in person a minimum number of hours per week during the entirety of your stay in Thailand.

Volunteering is illegal without a work permit, and volunteer visas were so abused for people trying to stay in Thailand that they're basically impossible to get anymore, so you won't be able to volunteer (legally).