The action button can be set to one of many functions, but volume up is not one of them.
If it's set to Ringer, it can move the volume from silent to ring.
Check Settings ➜ Action Button to verify what it is set to.
The other possibility is that since the volume up button is right under the action button, you are inadvertently activating the volume button at the same time as the action button. Though admittedly I find that kind of hard to do.
With that said screenshots use the Volume up button not the Action button, unless you have the action button set to a shortcut to take screen shots.
If using a case, it may be the case that is causing the unwanted press as mentioned. Remove the case and try again.