Yes, you can add active tasks to the AVI export.
TIP: Before opening the Export Animation window, create a Task Filter that shows only tasks that have 3D resources assigned. To do this, in Navigator>Task and Path Filters, add a new Task Filter. In the Generaltab below, select Resources. Open the Resources tab and select Mark All (You may wish to unmark any Human resources if applicable). Now the Gantt chart is only showing tasks that have a 3D resource assigned. When you add an "Active Filtered Tasks" text frame to the Animation (described next), it will not display tasks without objects assigned.
1. In the Export Animation window, open the Textual Frames tab. From the drop-down window choose:
2. After choosing the desired Text Frame, press Add New Frame3. You can update any of the Frame Properties including background color/transparency and font.4. After making updates, press Apply5. In the Content tab of the Animation Export, enable the new text frame by checking the box beside it, then use the arrows to move it on top of the other windows if necessary, and drag to resize and move in the preview
The final product will look like this: