Not always. Danish authorities handling CPR registration and residence permits generally accept birth certificates in English, while documents in many other languages need translating. You need a Danish translation when the body receiving your certificate asks for one, so confirm with them first.
Denmark is refreshingly bilingual about paperwork. Life in Denmark, the official guide for newcomers, says documents for CPR registration should be in Danish, English or German, and the residence schemes on nyidanmark.dk accept English too. So a UK birth certificate can often go in exactly as issued, which makes it the cheapest translation there is.
A Danish version earns its keep elsewhere: a municipality (kommune) or school that works only in Danish, a lawyer handling an inheritance or a family matter, a pension provider, or any body that has simply written 'på dansk' on its checklist. If you've been asked for Danish, that's the version to get.