If your REST APIs do updates or retrieve sensitive information then odds are you'll have some kind of authentication method, possibly with session cookies or whitelisted IPs. Allowing sensitive APIs to be sent via GET means that you open yourself up to a couple of easy vectors of attack. eg a malicious webpage with a <img src="http://example.com"> tag using the API end point as the src URI. This means the target person opening the page will send the API request as themselves, using their IP and cookies. Granted the API wont return an image so the "image" will fail to render but that doesn't matter as the API will still execute successfully. However if your APIs only accept POST requests then you mitigate this particularly attack.
It's pretty common advice to recommend any APIs that require user authentication to be sent via POST. In fact it's one of the first things pen testers will check for and you'd also fail PCI DSS vulnerability scans for exposing APIs via GET as well.
Disclaimer: I've works on multiple projects that have been pen tested, been audited by the UK Gambling Commission and/or had to adhere to PCI Data Security Standards.