Can a US Citizen for a US company work remotely in Portugal tax free while sightseeing?

In most countries, the tax code hasn't caught up yet with digital nomads and remote work, so it's a grey area.

A few things typically apply:

If you are a US citizen, you MUST file a US tax return no matter where you live and what you do. All other countries base taxation on residency, but the US (and Eritrea) base it on citizenship.
If you are on business travel, you are allowed to do work for your employer in some other countries without permit or local taxation unless you hit the residency threshold which is typically 180+ days per year.
What you are describing is a legal grey area that's not well defined.

That being said: for a month long trip, you should be fine, especially if you are reasonably discreet about it.