How long can I keep a permit open?

Check your coop or condos rules about this. My coop limited the length of a project, and would require you to stop work or issue a fine if work went over the maximum timeline. Just something to consider. Maybe others would know if the city has a time limit for the permit to remain valid. My guess is that there is an expiration date written on the permit though.

I believe typical NYC permits expire after 1 year but can be renewed (more fees...). You can check the expiration date(s) for yourself on BISWeb, using the DoB job #.

The building-rules aspect is probably more of a complication for your plan -- the management will, rightly, not allow your work to stretch on for months, even if it's only sporadically disruptive. To address that, you could plan the work in distinct, self-contained phases and plan to complete one well-defined phase per year. This will invite added scrutiny (vs completing all at once) but should be approvable depending on the details.