My job has always been remote, even before Covid. Would it be possible to live in Europe on a non-work visa and remotely work my normal job in America?
Regardless of work, residing in any country typically requires a special visa authorizing an indefinite long stay. Typically it's based on family ties, investment or work with the implicit wish of the lawmakers of allowing such for the sakes of taxes.
The only exception would be Svalbard https://travel.stackexchange.com/a/45131/4188
In the same thread, https://travel.stackexchange.com/a/89602/4188 points out that Canada allows you to work remotely during a visit, Americans can typically stay for six months in Canada.
Also, still the same thread, Barbados just launched a program tailored for you. https://travel.stackexchange.com/a/158462/4188
One addition to chx’s answer. During COVID, most of the European countries have implemented travel restrictions, where you are not allowed to enter the country unless you have a long-term residence permit from that country. As is, the rules for living in a country without a Visa are 90/180: so after 90 days, you are not allowed to stay in the same country, and must cross into a different country. This will be a problem if you do not have rights to enter that country.
That being said, if you restrict yourself to countries which do not have entry restrictions (e.g. Ukraine), and follow the 90/180 rule, then yes, you are allowed to do this.
Source: myself in the same situation.