Open source by itself isn't a tax-exempt action, but for many it's an important part of releasing free software for the public benefit. Services for the public benefit that are open to all can generally be framed as tax-exempt activity, but the lawyers at the IRS who review 501(c)(3) applications are tax attorneys, not software/tech specialists so a lot of confusion ensues.
Our review came back arguing that providing open source software is not only non-exempt, but actually provides a competitive advantage in the market because some companies would be paying for the same service, therefore they are at a disadvantage — even though they would have the same access to that free software as anyone.
Talking to other nonprofit directors and experts it seems like the arguments around open source are inconsistent at best. It makes sense because the IRS generally fast-tracks more common arguments and passes off specialty cases (think software) to individual reviewers. As enough case-law builds up they'll provide a more consistent decision based on internal protocols.
Out of the scope of your question, but the end result for us is that we're doing a bunch of legal wrangling, establishing more outreach and education efforts (which were always part of our mission) and getting ready to re-apply by year's end. It's not unrealistic to think a full ten years will pass from the start of the organization to the point where we get 501(c)(3) status.