If you said that the tone of the Epic of Gilgamesh was "serious," you wouldn't be far off. Most of the epic has to do with fights, quests, death, sorrow, and that sort of thing. This dark subject matter can definitely give the book a serious tone. Think about when Gilgamesh tells Enkidu, for example, that "as for human beings, their days are numbered, and whatever they keep trying to achieve is but wind!" (2.230-231).
Geez, Gilgamesh. Way to bring down the party.
But there are also plenty lighthearted moments, as when Enkidu first gets initiated into human society by Shamhat, and he drinks a ton of beer and gets a little saucy. Thus, we'd say that the tone of the Epic of Gilgamesh really depends on what is being described at any given moment: no matter what's happening, the poet simply gives you the facts: no fuss, no muss. How you emotionally react to those facts is up to you, the reader.