That certainly applies when stating an opinion.

If I say, "I believe that the Moon is made of green cheese", then it would be perfectly reasonable to expect someone to refute that argument by producing evidence to the contrary.

On the other hand when stating a 'fact' auch as -

"The Moon is made of green cheese", then it is up to me to provide evidence to support my statement otherwise it is perfectly reasonable for anyone to say, "prove it and if you can't then you are wrong".