"Spill" is also used for solid foods. In some circumstances, "drop" seems to be a better choice. For example:
Oops! I dropped my hamburger.
Dude, why are you so clumsy? You spilled all your food on my notebook.
From Cambridge Dictionary
Spill: to (cause to) flow, move, fall, or spread over the edge or outside the limits of something:
I spilled coffee on my silk shirt.
You've spilled something down your tie.
Let's see if I can pour the juice into the glass without spilling it.
He dropped a bag of sugar and it spilled all over the floor.
Crowds of fans spilled onto the field at the end of the game.
To cause or allow (a substance) to run or fall out of a container.
You can also say:
Be careful, don't spill your food.
Drop is when you are holding something in your hand and lose the grip, intentionally or unintentionally, and make the container or the contents to fall on the ground. Spill is more like an after-effect. When you drop something from your hand or accidentally tumble a sitting container and the contents pour out of the container or from your hand.
Suppose you are holding a basket full of popcorn. Now you drop it and what happens? The content, popcorn or the kernels, flows from the basket and are spilled across the floor. Mess is made when you spill something.
Take care; don't spill/drop the fries.
Take care; don't drop/spill the popcorn
Take care; don't drop the burger (burger is burger, it includes all the contents: patty, buns, sauce, tomato, onion, cheese, etc..)
Take care; don't let the burger fall apart.(I mean some pieces or inner content of it...maybe, some onion or sauce or anything that is 'inside' it).
...how to eat a burger without spilling the contents all over the table or down themselves.
I don't think any answer here would give a definite answer. But "drop" and "spill" are the most common verbs to define what you are looking for.
Like I said in my comment: the beauty of the English language is that you are not restricted to use only one word or one type of sentence to form your sentences.
Link to check:
Drop and Spill