YES, you can teach illegally in Taiwan, BUT you may be caught and expatriated immediately and your passport blacklisted for Taiwan return for up to 5 years.
NO, you can’t teach illegally because you do not have an Alien Resident Certificate (ARC) /work permit.
Teachers of English as a Second Language (ESL) are always in demand in Taiwan. With more than a thousand schools in the Taipei area alone, it is relatively easy to find work. There are still schools that need teachers badly, and will hire any teacher, with or without a proper degree and not go through the proper visa application process, with little fear of getting caught. In addition, during busy summer months, schools may add short term camps or summer classes that need part time teachers for just a few weeks.
Not too many years ago it was very easy to work illegally; however, those days are generally gone. The Taiwan government system has tightened up a whole lot in the last decade. Basically, you must have a work visa (ARC). The only way you can get a work visa is to be sponsored by the school that wants to hire you. The Education Department does make periodic checks on schools. If you are there and illegal when they arrive, it can spell trouble for you and the school you are working at.
Certainly you will still find some English teachers working in gray and illegal positions. Sometimes these positions are even good paying and very comfortable ones. These teachers have probably put together several illegal teaching jobs to earn what they need and make visa runs out of the country every three months. Back in the day, it was much easier to do this. Nowadays, by doing this too often, you may draw a red flag at a Taiwan visa office and be denied.
One final note, if you are illegal, you won’t have an ARC which means you won’t be able to open a bank account in Taiwan, get a full plan cell phone with a provider, or have an Internet account.
The ARC is your Taiwan ID card.