This is a very important question that not many people think about or consider. Tea boxes or tin cans are not an item that should be washed daily, maybe not even weekly or monthly, but with time, dust, little bugs and just particles that are in the air find their way into this places.
Since what you keep in this boxes is something that you eventually are going to drink, you should be very careful and clean with it. But do not stress, cleaning them is easier than you think, maybe even common sense.
If you box is made of wood or bamboo you can’t put it under the sink and wash it, just grab a damped cloth and clean your box inside and out, being very meticulous on the inside divisions and corners of your box or can. Do not clean it with any disinfectant or soap that might have a strong odor because when you add your tea into the box again it will be impregnated with that smell.