There are different rules applicable when a pet is travelling with its owner as a non-commercial move (e.g. for the purposes of a holiday) versus when a dog is being shipped commercially between Ireland and the UK (excluding Northern Ireland) (e.g. a dog being shipped to its new owner by a commercial transporter).
This answer details the rules for a pet non-commercial movement i.e. for an owner who may travel to the UK (excluding Northern Ireland) for a short stay with their pet and then return to Ireland,
Pet travel rules are found in Regulation (EU) 576/2013 and SI 141 of 2020.
In summary, to travel to the UK (excluding Northern Ireland), a pet dog travelling from Ireland currently requires:
• an EU pet passport,
• a microchip,
• a valid anti-rabies vaccination.
In summary, to return to Ireland after travel to the UK (excluding Northern Ireland), a pet dog requires:
• the same EU pet passport, microchip and rabies vaccination (provided all are still valid)
• a valid anti-tapeworm treatment certified by a vet between 24 and 120 hours prior to travel,
• to enter through a designated entry point - Dublin Airport or Port, Shannon Airport, Rosslare Europort, Cork Airport or Port at Ringaskiddy to facilitate a compliance check on arrival,
• the owner to notify the entry point in advance of their arrival to facilitate a compliance check (required for all non-EU countries).
DAFMs website provides a comprehensive breakdown of the above requirements in more detail if required.