Permits this? You are permitted to do anything you want unless there is something prohibiting it. You can certainly ground it if you want to as there is no prohibition on that.
NEC 392.60(A) applies to your question. The second part says "Metal cable trays containing only non-power conductors shall be electrically continuous through approved connections or the use of a bonding jumper." So I don't think it has to be grounded if it only has CL2/CL3 type conductors in it, just electrically continuous.
That being said, many would tend to ground it to something (an equipment ground, not a ground rod and not necessarily a ground electrode conductor) for liability reasons knowing that someone is likely to toss a power cable in that tray at some point. When you don't know the size of a possible future power cable that could energize the cable tray, many choose to run a #6 copper to it because that size seems to be the minimum where it can be exposed and not need much additional protection from damage. That #6 would come from the nearest panelboard or possibly building steel. If the cable tray is being supported by a strut rack and that rack is also supporting metal power circuit conduits, then I would deem it sufficiently bonded because of all the metal raceways attached to it.
