This is a complex issue with a couple moving parts, but I’ll try to be succinct. Especially in the social and natural sciences, PhD students are just typically funded. Lack of funding has a lot of direct and indirect negative effects.
If you are a student in a department where some students are funded and some aren’t, your lack of funding is (bluntly) a signal that you were a less competitive candidate than the students who did get funding. That creates a disparity - not only in available resources for students but also in how you are perceived by professors. Since advising, mentorship, and teaching by professors are key linchpins to PhD programs, that will have a negative direct effect.
Students who don’t have funding take longer to finish on average, probably because they usually have to work and/or take time off so they can scrape together funds to continue. They publish less, because if they have to work or otherwise worry about money, they are spending time they could’ve been using to do research and publish and present working. They probably have more extant stress, because every semester or year they’re worrying about how they’re going to secure funding to continue. That stress is going to interfere with being successful in the program.
And funding begets more funding: students who apply for fellowships and grants have to show a history of having been funded before. Ironically, if you already have a fellowship it’s easier for you to get another one. Conversely, if you aren’t funded, you can’t prove that other people think you’re good enough to give money to, so it’s harder for you to get new funding.
I was in a department where some students were funded and some weren’t. We had actual professors in the department telling students to turn down unfunded offers.
Well, first, Harvard (or any other elite program) isn’t going to accept you to a PhD program in psychology without funding you. They fund all their students. If they don’t want to fund your work, they reject you.
But at more mid-level or lower-ranked programs that accept some students without funding (or in fields where it’s more common for even top programs to admit some students without funding):
a) it really doesn’t work out that way. See above:
b) assuming you get funded, you still have to pay for the unfunded portion of your PhD. Even in a best case scenario where you get funding for year 2+, you’re still having to cover $60K yourself. If you have to cover two years, that’s $120K, which is more than the vast majority of PhD graduates can hope to make in their first few years out of grad school, which makes it difficult to repay.