It first appears in Mission 2, Burning Bridges. Charlie Jolson instructs Mark Hammond to go to The Republic and "trash the place to pieces". Mark visits the bar and kills every member of the Collins gang inside as well as shooting some lamps on the top floor which causes the restaurant to catch fire. Mark escapes the building just before it has completely caught fire and then he escapes the area. The bar is also seen in Mission 15, Disturbance in Soho. Frank Carter hears word that The Republic has caught fire and that there is major chaos on Frith Street. Frank arrives to find the bar completely trashed and on fire. Not only that, but members of the Collins gang have taken to the streets in anger and attempt to kill any police in the area. Once the area is safe and has been cleared of members of the Collins gang, firemen move in and put out the fire.
The Republic ultimately survives, as it is still the same as it was in the first game, as it is in The Getaway: Black Monday. This is further supported later on in the first game. If you drive down Frith Street in later missions, the bar can be seen perfectly normal and is how it was before Mark started the fire.