Probably depends on the technical definition of "disease" cus obviously stuff like cancer or vitamin deficiency (scurvy) are not infections, but I'm not sure if they're considered diseases.
Do you have a demonstration of the falsehood?
You just demonstrated the falsehood in your first paragraph; playing loosely with the definition of the word disease is precisely why the statement is flat out wrong.
but I'm not sure if they're considered diseases.
Which highlights the problem of just taking the platitude at face value. If you are not sure what is considered a disease, and what is not, then it is not reasonable to make claims about the platitude that All diseases are caused by infections.