I heard that a North Korean team played in the AFC Cup, which is a second-tier club competition in Asia. However, I have never seen a North Korean club playing in the AFC Champions League.
Is there any particular reason that they don't play in the Champions League?
It is not, as one might presume, because they did not qualify!
The AFC Champions League provides for the top twelve associations in each region to receive team entries, with the rating based on matches in the preceding four years at the time of assignment. For the 2022 ACL, this was the period 2016-2019 including the previous ACL and AFC Cup, up to 29 November 2019.
Korea DPR obtained just over 30 points and were thus ranked seventh in the East Region, twelfth overall in the AFC. This entitled them to "one direct slot", that is, to have one team start at the group stage of the tournament rather than play through qualifying stages.
However, the team that would be entered would declared ineligible as they do not have an AFC license; Korea DPR FA did not implement the license system requirements and therefore the entry is instead awarded to another team (details are irrelevant here), with the substituted qualifying entry also annulled due to requirements not met.
In other words, they could and should be, based on performance, but will not be, based on technical requirements not met.
Yes, they can.
The top-tier countries like South Korea or Iran have spots only in the AFC Champions League. Developing countries have spots only in the AFC Cup. Countries somewhere in the middle of them may have spots both in the Champions League and the AFC Cup.
The AFC distributes spots for the ACL and the AFC Cup separately for the east and west region. In each region, clubs from the top five associations have spots only in the ACL. From sixth to twelfth, they have spots both in the ACL and the AFC Cup. The clubs from the thirteenth and below associations go to the AFC Cup only. The rank of Korea DPR is 7th in the east for the 23-24 season and 9th for the 24-25 season. So they have spots in both competitions.
DPRK clubs didn't participate in the AFC club competitions in recent few years due to the COVID-19 issue. As another answer mentioned, they did not get the club license from the AFC, so it is not sure whether they would participate even after the COVID-19 pandemic.
As you mentioned, the North Korean clubs have never participated in the ACL but did in the AFC Cup.
Detailed regulations for the AFC Club Licensing can be found here. https://assets.the-afc.com/downloads/club-licensing/AFC-Club-Licensing-Regulations-2022.pdf
