One normally associates Beowulf (from the epic Beowulf) as an epic hero, not a tragic one. That said, when applying the characteristics of a tragic hero to Beowulf, one could support that he falls into this category as well.
In regards to his fight with the dragon, Beowulf desires to fight the beast alone. This could illustrate his hamartia (or tragic flaw) as one which is prideful and narcissistic). This said, his solo fight with the dragon could illustrate that he believes far too much in his own power that he cannot be beaten. This could show his tragic side.
That said, if one supports the idea that Beowulf fought the dragon for his people, this would show him aligning with the epic hero. One of the characteristics of the epic hero is to help others. By fighting and defeating the dragon, Beowulf insures that it will no longer torment his people.
Therefore, he would only be classified as a tragic hero if his hamartia was the reason he fought the dragon. Fighting for his people would not make him a tragic hero at all.