I will be hiring a car in late June , and have had no problems before and never been asked for the paper counterpart , but that is using the same company every year , and will be the same again this year !! I will still carry it as a precaution , but I'm assuming the paper details are now available online instead ?? Obviously I can get a code or view it myself , or the hire company can sign in and view it on the database considering I've supplied them with the personal details they need to view it , I did read a few weeks ago on a travel forum I frequently visit ( not TA !!! ) details about the change on June 8th and this link below also provided very useful information about the facts rather than just a journalistic view and the meltdown that could prevail

https://insidedvla.blog.gov.uk/2015/01/08/no-more-counterpart-date-confirmed-for-abolition/

I will report back if any difference in my experience this year when hiring , when I get back !! But it has helped as a useful reminder to get my wife to renew her licence online for the newer photo licence ( they drag your photo from your passport ) as she still has the old pre 98 paper ID , mine was a tattered torn and eight piece document which became an embarrassment when presenting it at Lanzarote airport each year

It'll be interesting for the first few months -we hired a car from "Costas" (or similar!! ) when in Corfu 18 months ago, and he certainly copied my details from the paper counterpart - I've also been asked for it in Rhodes in 2012 and they wouldn't add my wife because she still has the old paper licence (no photocard). There is no way that they are going to download stuff from the DVLA site and I'm not sure how much use the "code"is going to be,especially as its only valid for 3 days !!
My S-I-L flew to Malaga yesterday and picked up a car -I'll have to ask him whether he had any problems in advance of the changes.

EDIT Just tried to log in DVLA site to view my licence -website was down ! Good start !!