This can go either way. Silver bordered card rulings are complicated (see the caveats below) and Mark Rosewater has issued conflicting rulings on this.
Mark Rosewater's Rulings
MaRo's original ruling on the subject is that the owner of Ashnod's Coupon still pays for the drink:
R&D Secret Lair contradicts itself as it asks you to play “all cards as written” and “ignore all errata”. Ashnod’s coupons errata is written on the card.
This means I can basically rule either way so I have opted to rule in favor against dumb shenanigans. (Note I’m all for regular shenanigans when playing with Un-cards.) Owner of the Ashnod’s Coupon pays for the drink.
MaRo's later ruling on the subject (thank you @Malco for finding this) is that the other player pays for the drink:
If you use ashnod's coupon and you have R&D's secret lab out, who pays for the drink?
They do. Note that they can concede in response to the activation.
Silver Bordered Card Rulings
Silver bordered cards are complicated. The official magic rules say the following:
Cards in certain sets and certain promotional cards are printed with a silver border. Silver-bordered cards are intended for casual play and may have features and text that aren't covered by these rules.
Wizards does not publish official rulings on silver bordered cards the same way they do on regular cards. Mark Rosewater (MaRo) is Magic's "Silver Bordered Rules Manager" as mentioned in the FAQs for Unhinged and Unstable, the later of which lists MaRo's blog as a resource for answers to additional questions on silver bordered sets. While these are official publications of Wizards, they are not included in the regular rules cannon, and so absolutely do not have the same force of authority as the main magic rules or card rulings.
MaRo has said:
One of the wonderful things about silver-bordered cards is that we can do stuff that maybe doesn't technically work but that players would have lots of fun trying to make work.
Silver bordered cards were never intended to have a coherent rules framework. This is visible in MaRo's own discussion of this situation, where he describes the iteration between these two cards as "contradictory" given the magic rules framework, and his ruling was based on a personal philosophy rather than any logic of the rules.
My Personal Opinion
I believe that the targeted player pays for the drink.
Even in MaRo's original ruling, he claimed that the interaction could go either way. "Ignore all Errata" is its own sentence, and thus can function as an independent part of the ability: first, play all cards as written, and second, ignore all errata. Under this interpretation, the errata on Ashnod's Coupon is nullified by R&D's Secret Lair, and the target player will have to pay for the drink.
In this case (as highlighted by MaRo's later ruling), a player can still conceded the game at any time. Thus, this combo, targeting a sufficiently expensive drink (such as a bottle of 50-year Macallan, retail price $30,000), effectively becomes "Target player loses the game". This makes it a 2-card 0-mana win combo, which is the simplest first turn win combo in all of magic. The downside (?) to this combo is that a player does have the option to buy you a $30,000 bottle of scotch to prevent themselves from losing.