Almost everyone in Denmark can speak very good English. You are unlikely to have many problems communicating with the Danish people, so learning to speak Danish is not essential. German is also widely spoken in Denmark.
Despite this, being able to speak at least some Danish may be helpful at some point during your stay in Denmark. Understanding the local language and mentality will make it easier for you to immerse yourself into the Danish lifestyle.
Many international companies operating in Denmark use English as their main business language. However, some smaller firms do operate in Danish, so some knowledge of the Danish language may be useful if you plan to apply for a job in Denmark.
You do not have to speak Danish in order to study in Denmark. There are intensive language courses taught at higher education institutions for those who do wish to learn Danish (see below); however there is a wide range of courses taught in English.
If you try to speak Danish, it is difficult to say whether the locals will appreciate your efforts or not. Some Danish people certainly will, but at the same time it is not unusual for Danish people to answer a question in English, even if you ask in Danish!