How important are referrals when applying for a job as a developer?

Important, but not as important as everything else. Keep a few references. Let your references know you're using them as such - I feel like doing so is considered a best practice. Don't sweat it. If you're good to people and generally a good person to work with that cares, your references will be a good asset to have.

Also, be happy to be a reference for someone else. You can change someone's life by being a good reference.

Important enough that you shouldn't give them to a recruiter until the offer stage of the interview process. Many staffing firms are now doing preemptive reference checks. This is problematic for potentially two reasons. Even if the first doesn't apply, pay attention to the second:

1. These early reference checks are often a cover for unscrupulous lead gen tactics by these agencies (they're often conducted by account managers who are more interested in drumming up new business clients than they actually care about checking your bona fides).
2. (again, even if 1 doesn't apply) They exponentially increase your references' exposure to such calls. You depend on these people, who are doing you a favor, so you need to protect their time even if staffing agencies don't respect it. The industry-standard etiquette for decades has been to check references at the offer stage of the hiring process, so references have (reasonably) had the expectation that they would get one call (maybe two, absolute most, if the first falls through at the last minute) per job hunt, not per job applied to.

People often wonder why I stress this so much. It's because I lost two of my best references when this awful trend started, precisely due to overexposure. I had no choice but to accept their reluctance to provide future references and apologize for something that wasn't my doing (calling a reference, in some cases, before I had even been submitted for a role, let alone before an interview had been conducted).

As I said before, you depend on these people and they're doing you a favor. Protect them from overexposure.

Sometimes they're important, sometimes they're not. I've never been hired in a developer role based on a referral, but I've been on teams where large numbers of employees were referred by people who were already working at the company.

Like most factors in the mystical world of hiring, your mileage may vary.

IMO: Not important. Referrals always seemed like a box-ticking exercise in the process. As long as you have some, you'll probably be fine.