I'm not going to lie, I've taught in China part-time on tourist, business, and student visas. This practice is pretty common, especially in the PRC, but ideally, you should be on a legal visa.
Why? Well, if you're going to work abroad it's probably better if you're doing so legally. This is especially important if teaching is your full-time job.
Some countries like China are notorious for hiring full-time teachers on tourist or business visas. This is because the demand far outweighs the supply of qualified English teachers.
If you're not a native speaker or you don't have a degree, it's almost impossible to get a legal visa in most cases, so working on a business, student, or tourist visa may be worth the risk.
But if you do meet all of the qualifications, there is absolutely no reason why you should take a job that doesn't provide a legal visa for you. You should receive your visa either before you arrive or immediately after you finish training. Otherwise, you should start looking into different jobs.
If you have the qualifications, you deserve to work for a company that hires you legally. Period.
How Do You Know if You Have the Right Visa?
Don't immediately trust that your employer will hire you on the appropriate visa.
Do some research online to learn what type of visa or permits you need to teach legally in your country. If there are discrepancies, ask your employer to explain. If you're still not convinced, you can ask a visa service agency, or a relevant site online and compare their answer with what your future employers tell you.
When in doubt, just find another position. Better safe than sorry!