Only over some threshold (iirc, though it has been some time, around 80k USD equivalent). So maybe yes for someone with a comparatively high income, but not so much a worry for a reasonably large segment of the population. Naturally if this happens to apply to anyone reading I would urge consulting a professional for current advice. ;)

The US has tax treaties with most developed countries, including Finland.

You still need to file tax returns, even if you are under the limit to prove that you owe no tax.