If you need a non-geographic number, you'll need a number starting 03.
If you are starting out with a brand new number, you'll need a number starting 033.
If you already have a number starting 084 or 087 it can be changed to the matching number starting 034 or 037.
Calls to 03 numbers are inclusive from landlines and from mobiles, else charged at geographic rate, the same as for calls to geographic 01 and 02 numbers.
The non-geographic number company will need to be paid for handling and forwarding the calls. If you shop around you should be able to find deals where you pay a small fraction of a penny per minute for incoming calls.
If you choose a number starting 084, 087 or 09 (don't do this!), you force those costs onto callers, and (since 1 July 2015) you have to declare this Service Charge in close proximity to the number everywhere the number is advertised or promoted. With this type of number (084, 087 or 09), everyone else in the chain also adds their margin, really ramping up the cost to callers. Since 13 June 2014, it is not permitted to use this type of number for post-sales helplines, see Regulation 41 of the Consumer Contracts (Information, Cancellation and Additional Charges) Regulations 2013. Additionally, usage of 084, 087 and 09 numbers for sales and pre-sales enquiry lines is a well-known 'sales prevention technique'.
Do also have a read of
http://www.fairtelecoms.org.uk/service-numbers-08-09-and-118.html and the pages it links to.
Additionally, do not be tempted to use numbers starting 055, 056 or 070. These have weird call charges and most consumers distrust them so much they would go out of their way to avoid calling them.
Ireland has a big problem with 1850, 1890 and various other such numbers. The EU Consumer Rights Directive was a massive opportunity to put things right, but the Irish Government really messed it up and consumers have not gained the protection that was intended by the EU. I posted some background information on various .ie consumer forums at the time, but there was little interest or response.