Yes Dan, as long as the tab is open you will still get information from the device.

You can actually see this in my Web Bluetooth talk from Jazoon Tech Days:

at 33:33 I ask a volunteer from the audience to connect to my Purple Eye robot from his phone using Web Bluetooth, and then you can see the Robot dancing as the Chrome tab is sending instructions to the robot to move its motors. Then, I ask the volunteer to lock the device screen (which is essentially the same as moving to a different tab / switching to a different app), and you can see that the robot still moves, which means the connection is still alive and functioning.

However, you will notice that the robot has slowed down. The reason for that is Chrome throttles timers for inactive tabs, so the timer only fires once a second (instead of every few ms).

I hope this helps!