Cognitive screening has been proposed for older adults diagnosed with chronic obstructive pulmonary disease (COPD). Therefore, we examined the change over time in cognitive function and the risk of incident dementia in older adults after COPD diagnosis. A sample of 3,982 participants from the population-based cohort study Good Aging in Skåne was followed for 19 years, and 317 incident COPD cases were identified. The cognitive domains of episodic memory, executive function, and language were assessed using neuropsychological tests. Mixed models for repeated measures and a Cox model were implemented. Participants performed, on average, worse over time on all neuropsychological tests after COPD diagnosis in comparison to those without COPD, although statistical significance differences were only observed for episodic memory and language. The groups had a comparable risk of developing dementia. In conclusion, our results indicate that cognitive screening in the early stages of COPD may be of limited clinical relevance.