How do I install ios10 without a developer account?
Without any research on the matter I would say: you can't. 

That's exactly what developer account are for: so a new software / a new system can be released to a small group of expert users. While the software / system may already look ok in a presentation, it probably still is way too buggy for the average user. So  either be patient, or become a developer. 

Also, Apple has released the last few major OS upgrades (for both Macs and iPhones/iPads) as Public Beta some months before the official release. This will allow you do get your hands on these new versions some time between now and whenever-the-release-date-will-be.