During that time of year, the connection Hamburg–Copenhagen is very busy. Before they introduced compulsory reservation in the summer months, I had a train that was so full, I had to almost physically fight my way to my resered seat, and this almost happened. (They did leave in the end — with a delay — but still very packed.)
Thus, you should reserve a seat at the earliest possible moment (i.e. once you have decided on a connection). This is easiest via bahn.de, which has an option in the buy tickets dialogue to only buy a reservation. It is very likely all the seats will sell out at one point (especially for the trains that are not the first or last on a given date) but I can’t predict when that will be.
The next time when I travelled on it (summer 2014), DB security service appeared on the platform in Hamburg before the train entered. Once the train came to a stop, each door was guarded by one or two of these security officials, whose job it was to make sure nobody boarded the train without a valid reservation. Thus, I very much assume that the answer is no — they won’t even let you board without a reservation. (That year, the travel was much more comfortable — you could actually walk through the corridors.)
You could attempt to circumvent this by taking a regional train to Lübeck and boarding the ICE to Copenhagen there. I can’t remember whether DB security service was on that station, too. (Since the next stops — Oldenburg (Holst) and Puttgarden — are very inconvenient to get to, that is even less an option.)
Note that once you crossed over into Denmark, a second unit is typically added to the train (in Nykøbing Falster, iirc). Since many people prefer to stay on their reserved seats and not move their luggage (in spite of many announcements by the train crew that an almost empty unit has been added), you should find lots of free space from then onwards. The Danish leg has no compulsory reservation, since space is no longer a concern. (Danish single tickets come with free reservation afaik, though.)
Note that this answer is written concerning the ICE trains Hamburg–Copenhagen. The situation will be different on other trains with compulsory reservation such as night trains (CNL and EN until December, only EN after that) or other high-speed trains (e.g. Thalys).