iPhone Will not Auto Lock

First, make sure auto lock is enabled in settings. I personally do not use the feature and lock my device manually whenever I am done with it, but that is a personal preference.

Don't be afraid to press the power button. You will not lose VoiceOver, regardless.

If the device does not lock, even when you press the power button, try restarting the phone. Hold down the power button until you get the option to power off. Double-tap the screen, then wait several seconds for the phone to power down. Then, press and hold the power button for a second or two to start it up again. It will take a while before the device is up and running again, but it shouldn't be more than a minute. You will need to enter your pass code instead of touch ID, but other than that all should be well.

Again, at no point should you lose VoiceOver, because that setting is sticky and is never lost, short of a complete wipe of the phone.

Agree with previous commenter. I never use auto lock on my devices. I find it annoying and inconvenient. However, you shouldn't be afraid of using the power button, you won't loose Voiceover at all, it just locks the phone, and when you hit the button it'll wake up the device and VO usually starts speaking shortly after.