燃えるゴミと燃えないゴミは分別して捨ててくださいって英語でなんて言うの？

It's important to stress that this is a rule in Japan, because most people are not accustomed to separating their trash.

You could also tell them: "In Japan we separate our burnable and non-burnable trash, and if we don't do it sometimes someone will bring the trash back to our houses."
If you explain the consequence they will take it more seriously.

Combustible = capable of catching fire and burning; inflammable; flammable:
"Gasoline vapor is highly combustible."
Flammable = easily set on fire.
Non flammable = not catching fire easily.
"A safe, non-flammable adhesive."
"John, please separate flammable things like paper from non flammable things like metal."

flammable items - these are items which can be easily set on fire; materials that burst into flames easily

In the US, we usually don't separate items in this way, but we do recycle: this means separate the non-renewable/non-recyclable food waste "garbage" and the renewable/recyclable plastics, papers, and metals.

Some places have a "single stream" recycling system, meaning that all recyclable materials go into one bin/can. Other places have "double stream" or "triple stream" or completely "non-integrated" where plastics, metals, papers, etc. need to be sorted into separate cans/bins. "Single stream" is very useful and makes recycling so much easier!

"You need to sort the trash/rubbish." This explains that you need to sort the trash you cant just through it all in the same place.

"Trash" and "Rubbish" have the same meaning as "Garbage".

"You need to separate the rubbish." This is another way of saying you have to sort the rubbish.

"You need to recycle" In the UK sorting the rubbish is called recycling.

This will help them understand which type of trash goes where. It is clear and also comes across as polite especially if you say, "if you don't mind...".

"You must separate the trash" A statement implying that you must separate/divide the trash in to correct bins, In this case Burnable and nonburnable.
"Trash must be segregated" To 'segregate' something is implying that it must be separated/divided.

Recycle is a verb that means convert/change (waste) into reusable material.
Therefore is an adjective that means material that can be reprocessed to something that can be reused e.g tins.
Non recyclable means not able to be processed or treated for reuse in some form. For example paper

Using the above would be a good way of showing them that the trash needs to be separated into different bins depending on the type.

1. Please put the burnable trash in one bin, and the unburnable trash in another bin.
   This is a simple and direct way of asking someone to separate the burnable and unburnable trash.

In many countries people have become aware of the need to recycle wisely...
Indeed some places will now "fine you" (charge a fee) if you fail to separate your rubbish properly! It may be there are now bins for different materials such as:
Glass... Tin...Carboard...Plastics...or indeed Flammable and non flammable!
Recycle well!