Does iOS Health app send your health data to Apple? I am wondering if the biometric information collected by Apple's Health app stays local on my iPhone, or if it's being sent to Apple?
Unless you choose to backup your data to iCloud, none of your data will be sent to Apple. Even if you do choose to backup your data to iCloud, that data is encrypted - meaning only you can access it.

Apple’s privacy policy is here and you can request Apple send you all the data they have and inspect the health data yourself to know how / when / what was collected based on how you set up your accounts and sync options combined with how iOS is programmed.


https://www.apple.com/legal/privacy/en-ww/ (Read about policies and related links)
http://privacy.apple.com/ (Sign in and get your data)


However, you also decide what 3rd party apps, if any, you want to have access to your health data. These can be set via the Health app directly or via Settings > Privacy > Health. So make sure you check their privacy policy if this concerns you!

To quote Apple directly: 


  The Health app lets you keep all your health and fitness information under your control and in one place on your device. You decide which information is placed in Health and which apps can access your data through the Health app. When your phone is locked with a passcode or Touch ID, all of your health and fitness data in the Health app — other than what you’ve added to your Medical ID emergency card — is encrypted with your passcode. You can back up data stored in the Health app to iCloud, where it is encrypted while in transit and at rest. Apps that access HealthKit are required to have a privacy policy, so make sure you review these policies before providing apps with access to your health and fitness data.


Source: Apple 

In a nutshell, only you (or anyone who knows your passcode or has a fingerprint set on your iPhone) have control of your health data.