I had this identical issue for months until I got totally fed up with the workaround (calling my voicemailbox for messages - the "old" way) and finally sought out this forum thread yesterday. This was impacting my business and I'm sure it's equally frustrating for others too. I called Verizon and had a Level 1 Tech explain to me how the integration of Visual Voicemail into iOS 10 had taken place. Long story short - after disabling and re-enabling the Visual Voicemail Feature a few times (and power-cycling the phone after each individual toggle of that feature) all of the voicemail transcriptions now remain open when each one is touched. And I can actually listen to the voicemails without the transcription collapsing and stopping the playback, which was even more annoying. For me, I could live without the poor quality transcriptions but not being able to playback the message right on the device was a frustration that broke this camel's back.
To be clear on how I resolved this, while logged into MyVerizon, I disabled the Visual Voicemail feature and hit "save" to confirm its association from my device at the account level, restarted my iPhone 7, then repeated the process by re-enabling the feature on MyVerizon, saving to the account again, and then recycling the iPhone power once more. To be sure, I waited 10 minutes between each change and power recycle to ensure the network was aware of the change and had a chance to push that change down to my device.
So far the remedy seems to have "stuck" for me and I recommend others give it a try as I know the joy firsthand of having this feature function the way it is intended. And now for my cynicism. If this by chance ends up being a short-lived solution, rest assured I'll back on this forum in a heartbeat letting you all know and picking my torch back up.