The primary feature that is affected is feature spacing. Due to the effect of lateral etching, PCB fabricators must apply an "etch factor" or "etch compensation" to the customer's design features. Typically we add 0.001" to each side of a feature for each ounce of base copper. For standard 0.001" plating in the holes, the base copper would be the required finished copper less 1 ounce. You may find our heavy copper spacing calculator to be handy.
The other features that are affected are the dielectric thicknesses between copper layers. In order to prevent resin starvation, or voids, between layers of heavy copper circuitry, the PCB fabricator must insert more plies of prepreg (bonding resin) than what is normally considered standard. As such, avoid calling out too thin of an overall thickness or dielectric thickness between opposing heavy copper internal layers.
Given that we are PCB nerdy, we love helping customers out with stackup calcs on heavy copper PCB designs. Just give us a shout.