When you apply for citizenship, you check a box for them to pull your entry and exit stamps. How long does it add on ircc's end for them to do so ?
IRCC will examine the applicant's passports for entry and exit stamps.
Item 15.b. in the June 2019 version of the citizenship application has a box to check giving CBSA permission to "disclose the details of [the applicant's] history of travel to IRCC." This is not about "stamps."
IRCC can access the CBSA travel history electronically, so the amount of time allocated to do this particular task is probably a small part of an hour.
As with almost every aspect of the citizenship application process, how a particular task affects the processing timeline is far, far more about whether there is a separate queue for that task and if so, how much longer will that queue add to the process. But the other big time factor is whether or not in reviewing the travel history IRCC sees reason to engage in RQ-related non-routine processing. It is the latter which can really add a lot of time to the overall processing timeline (albeit, again the vast majority of that additional time is queue time).
It is not clear whether IRCC accesses the CBSA travel history for all applicants. If IRCC examines the CBSA travel history for all applicants, there is no separate queue outside the routine process, so for the vast majority of applicants this will add NO additional processing time at all.
Not giving consent for CBSA to share your travel history with IRCC is much MORE likely to lead to some delay. And there is no reason to not give consent. IRCC can still access the information, but without consent it is a more involved process to comply with privacy safeguards and information sharing protocols . . . so they can still get it but that would almost certainly mean an additional non-routine process with a separate and probably lengthy queue of its own.
These days most PRs, including those applying for citizenship, do not have exit and entry stamps in their passports for every trip. The CBSA travel history data is a far more complete and accurate source of at least entry dates (while those records are highly accurate, they are not necessarily complete). Again, item 15.b. in the citizenship application is about CBSA travel history and NOT about entry or exit "stamps."