System Information
Operating system: MacOS 11.2.2
Graphics card: Intel Iris Plus Graphics 640

Blender Version
Broken: 2.92
Worked: 2.91.2

I have created a Grease Pencil rig that uses Time Offset modifiers. It functions correctly in the viewport window, but during render it plays through the frames as though there was no modifier. I have not tried this in Cycles yet. It appears to work fine in 2.91.2, so this is a new bug.

I've attached a WIP of the rig, the (unfinished) body frames are supposed to remain the same, and the head is supposed to move with along with the face.

Someone reported a similar problem on this link while using the time offset on loop: https://blenderartists.org/t/loop-only-plays-once-when-rendered-help/1288262

To create this, create a Grease Pencil object with multiple frames, and then put a time offset modifier on it as either fixed frame, or a loop. Test to make sure the modifier is working correctly in viewport, then attempt a render.