For a short stay in Japan (90 days or less) no visa is needed, only proof that the stay will be shorter than 90 days is required.
During this time, work in Japan is not permitted, but what about working for an online, American company, receiving payment from the company to an American bank?
Is there a limit to the number of times I can leave and re-enter Japan, thus effectively resetting the 90 day limit?
The bigger limit you'll eventually run into is the half-year limit. If you're staying more than 50% of the year within a country, it becomes your primary place of residence, which entails a bunch of rules. Most notably, you must be taxed in your primary country of residence, which is somewhat impossible if you're there as a tourist. So, you cannot spend more than about 180 days per year on a tourist/temporary visa in Japan.
Constantly coming and going shouldn't be an issue, as long as you stay under the 50% limit. It might raise some eyebrows if you're indeed constantly coming and going, but technically there's nothing anyone would do about it besides perhaps asking some more questions. Practically speaking, if you're coming and going often enough within half a year to raise eyebrows, you're probably spending a fortune on plane tickets; and if you have enough money to do that, you are probably in a position to get some actual visa.
Personally I know of "tourists" who did the "Korea run" a few times; flying over to Korea to come back on a new visa. They always maxed out their 90 day visas, they didn't go back and forth enough to test out any limits. I know that in some cases, they were allowed back in a third time (270 days total), but with a stern warning, and never a fourth time (I don't think anyone was dumb enough to try). Such exceptions are entirely at the discretion of the immigration officers, who will judge you on your individual situation, and it's certainly not something you should expect or push your luck on. It's also been a number of years since, I don't know what the current climate is like; they may be much stricter these days.
I did hear of someone who was dumb enough to declare he was working (in Japan, for a Japanese company) while applying for a tourist visa; he was send back home immediately...
For Japan (And many countries):
It depends. Lots of people do it, but technically your visa is for tourist purposes, and you're meant to be doing that, not working. If you're working during your stay, you're arguably violating that. It's a grey area, as a lot of people do it (See people working in Chiang Mai, Thailand).
In theory, no. However, it's going to start raising questions - they'll wonder if you're trying to 'live' there or 'work' there without getting residence / avoiding tax. At which point they might start wanting evidence of a job back home, or that you have a home elsewhere and that you do plan on leaving eventually. Same with most countries.
