Looking at the specification for it
http://www.pcadvisor.co.uk/new-product/digital-home/amazon-fire-tv-stick-release-date-price-specs-uk-cut-3582765
I'd expect services that aren't restricted to the UK (irrespective of the device you're using to attempt to view them) to work normally but those which are 'UK only' won't.
So, for example, you'd be able to view exactly the same Youtube content as you would via a PC (which means most of it but with some region-specific content not available). The same would apply to Daily Motion and Vimeo.
Netflix is available in Ireland but the app on the Amazon Fire stick might be designed to connect only to the UK service, so you could have a problem there.
UK catch-up services (such as BBC iPlayer) almost certainly won't be available in Ireland because the BBC's servers will see that you've got an IP address located in the Republic.