Does Conkeldurr's "Chip Away" attack ignore effects such as Weakness and Resistance?
No, Weakness and Resistance are not considered effects; they are basic game mechanics and are not ignored by Chip Away.
Source: Noble Victories FAQ (2011-11-17), PUI Rules Team (2011-11-17)

Relevant cards

Note: All applicable cards may not be listed below.
BW3_65