You are talking about complete finished consumer products (e.g. bottled jam)?
This actually happens a lot. A lot of retailers also require you to store your product at a third party warehouse of their choosing (and off course they only pay for the product once they have shipped it out) .
You should do a risk assesment (as always ) but since your packaged product is low risk (long shelf life, no refrigeration needed etc.) there are not that many things that could go wrong. Check with your own CB is if there are additional requirements.
You should at least have a signed statement of the 3th party that they comply to all relevant food safety regulations (e.g. proper storage, pest control, FIFO, cleaning, basic employee knowledge that food is handled). And performing audits/inspections at the side is a smart way to create control.
I agree with this.
Regarding the retailer requirements, I know there are some retailers that require 3PLs to have GFSI (just to warehouse finished goods). I understand that some products carry higher risk than others, but I know at least one that doesn't risk assess. They just require everything to be stored in a GFSI certified warehouse whether food or non-food, with no thought to risk.
In this case, the OP would have to work with their team to identify a warehouse that meets this criteria.
Alternatively, if this is to satisfy GFSI requirements internally, the warehouse does not need to be GFSI certified as mentioned by Scampi and Marloes. Just need to be sure to conduct the proper assessments to meet the standard.