Aruba says tourists are not allowed to work there. How strictly should I interpret this?

There are two issues here:

1. "not allowed to work"

This refers (normally) to taking [illegal] employment in-country on a tourist visa. And thus avoiding the whole employment visa process, and paying taxes. FYI, here's how Aruba defines work.

Generally speaking, Immigration doesn't care about you checking your work email. They're usually more concerned about illegal work and immigration.

2. "an excuse to turn off while I'm there"

You don't need an excuse. You're on vacation. Depending on where you're from, pestering you to check your emails and do some work could be a (serious) violation of your country's laws. If you want to turn off while on vacation, leave your work phone and laptop at home.

Using I want to use this as an excuse to "turn off" while I'm there. is beyond ridiculous. Your manager would probably laugh at you so hard your ears would ring even after your vacation is over. Don't. If you need an excuse to not answer emails, you need to find another job.

De jure there are dozens of complex laws and regulations regarding remote employment, under which you may or may not need a special visa in order to work in a given country. Tax laws are an additional complication, where countries such as the UK can deem you as a tax resident for spending as little as 16 days on British soil.

De facto, as long as you don't mention your remote job to immigration personnel at the airport, there's a 99.99% chance no one will ever find out. There are millions of people breaking the law by being employed at on-site jobs in any given country, so remote workers who travel for short periods of time are a pretty low priority for law enforcement.