The general rule is "if you make money while in a country, you need a work visa while in that country", but obviously that may be a bit outdated and not take into account remote positions. Many countries can/will fine, blacklist, and deport you if they find out you worked while within the country without a work visa; and it's _those_ people that you need to be able to convince "I'm not taking your jobs away!" to.
I'm a US citizen so I haven't looked into it from your POV, but I do know that a trip to Canada, for example, allows working remotely as long as:
* you work for an American company with no Canadian offices or branches[1],
* you are paid in American dollars, and
* that payment is deposited into an American bank account.
Again, depends on the country you're visiting (and possibly the country you're a resident of), so I'd double check before working on a trip just to be safe.
[1] If your company has branches where you're going, the safest bet is to get them to help arrange the trip. A have a family member that travels a lot on business and has been denied entry to countries while on business trips specifically to train the local branch, because even then it's a delicate situation.