Depends on what the separator sheet is made of.
If it's wax paper sheets, it will likely turn to pulp from all of the moisture and kill your burgers based on what happened when I just went and dunked a Weston wax paper burger sheet into water, let it sit a minute then moved it around and rubbed it gently. Turned into complete pulp.
Now, if it's cooking parchment paper, that likely would be fine. There's an entire style of cooking, known as "En Papillote", where the food is sealed up in folded bakers parchment paper and allowed to steam in it's own moisture. It's pretty durable stuff.