I fail to imagine any situation where a non-profit project could fail because its sourcecode is non-free. But I will give you the benefit of a doubt and assume that it somehow is.
It's sometimes surprising how much time and effort people are willing to invest into projects they believe in. However, when you make your project closed source, you are erecting some additional barriers for new volunteers:
- People are shy. They won't ask if they can help you when they aren't sure they actually can help. When someone notices a bug or misses a feature and the sourcecode is available, they can check it out to see if they understand it well enough to help you. But without seeing the source beforehand, they won't be so sure if they can actually do it and will be more reluctant to offer help.
Open Source greatly reduces onboarding time for new volunteers. In a closed-source project you need to get into contact with them, you need to give them access to your version control system and you somehow need to convince them to not leak your sourcecode to the public. Volunteer initiatives of any kind usually have a very high turnover rate, so this administrative overhead might become considerable.
In an open source project they just need to do a checkout of your public repository. With websites like Github they can branch your project with one click and send you a pull request with another.
- Without an open source license, people might be afraid that you will one day turn your project into a for-profit project and exploit their hard labor without paying them.
Oh, and by the way: "Open Source" and "For Profit" are not necessarily contradictory. See How can large open source projects be monetized? for details. In my opinion community-driven websites are perfect for monetizing as open source project, because the asset with the most commercial value is usually their user community, not their sourcecode. People can't fork the code and expect the community to follow.