Burgers would be potentially unsafe only because of a shared cooking surface.  I don't really worry about the seafood cross contamination unless it would be blatant because they are now testing negative.  Same thing for shared fryers for fries.  I would want to check that there are no explicit nut items on the menu.  I was just at a restaurant last month where my friend got sick (Celiac's) due to a shared frier.  I also found a bean in my dish that had clearly dropped off someone else's plate when it was lifted over mine. 

Just checking...yes I would not use ice cream made on shared lines with almonds.  We are PA and TNA only (well pretty much). 

This is regarding Disney World.  We've been there 3x and eaten successfully in the restaurants.  However, they emailed me a list of safe counter service items and it included rice dream ice cream.  There was very little on the list which had me irritated because I would like to plan where to eat ahead of time, but I will post that elsewhere

Brownie

For nuts I'd almost got for plain Breyer's vanilla over Rice Dream. I eat it and my containers have warnings, but even if they didn't I think all their lines are shared with almond.

Is it the seafood that makes the burgers unsafe?

Rice Dream has cross contamination warning on it:

QuoteManufactured in a plant that uses peanuts, milk, eggs, wheat and tree nuts. Good manufacturing practices are used to prevent introduction of these or any other unlabelled allergens in the product.

I don't know which Disney list you are looking at.  I've found Disney chefs very strict on not serving things with may contain warnings on them.  Even if there is a list, always ask for the chef, mention your allergy, and double check ingredients before ordering.

Rice Dream has or has had some cross-contamination issues.  I don't know about peanuts but def with milk. 

Which Disney? And are you looking at more than nut-safe things? Because when I was there there was more things that I COULD eat than things I couldn't...

Is this actually pnut/tnut safe? Disney has it on their list of nut-safe foods.  Considering how limited this list is, I am surprised bc we have never eaten this.  There is practically nothing on the list besides allergen free chicken tenders so I am a little irritated by the whole thing.  We've eaten at the restaurants there and yet there is not a safe burger on the list (but there is a safe bun! that's helpful without the burger - not!)
Brownie