Can I say 'it has been and will continue to rain' or should I say 'it has been raining and will continue to rain'
You would have to say "It has been raining and will continue to rain".
"It has been and will continue to rain" would be forgiven as an error in casual speech, but is not really grammatical, because the required form ("raining") is different from "to rain".
The fact that two verbs govern the same participle isn't itself the problem: "He should and will leave" would be grammatical, and so would "He should have and will have left". Rather, the problem with your original sentence is that only "has been" takes a participle, while "will continue" is followed by the infinitive.
"It has been raining and will continue to rain" emphasises the fact it's been raining. But if you just said "It will continue to rain", that would mean essentially the same thing, since it can only continue to rain if it has been raining.