For example, there are biological science and scientific news. Can I say it 'biologically scientific news'?
In this context, 'biological' means 'relating to biology,' and 'scientific' means 'relating to science.'
Short answer, no.
There are situations where you can put an adverb, adjective and noun together, for example "statistically significant data".
Biological science, however, is a compound noun, made up of an adjective and a noun. You can't just convert each of the parts to something else, and expect it to be a "compound adjective" or whatever.
Fortunately, there is no limit to the number of words in a compound noun, so it's perfectly OK to just tack another noun on the end. In this case, you get "biological science news".