We explore in some detail why this book falls into satire category in the "Genre" section, so you probably want to read that before we get going with tone. We're happy to wait while you hop on over there.
Okay—back? Let's go.
A good way to detect a satirical tone is to keep your eye out for overblown scenarios and language. Just consider Hank's commentary on the armor knights wear. When he says, "A man that is packed away like that is a nut that isn't worth the cracking" (11.9), we understand that not only does he find the armor itself ridiculous—it packs its wearer away, obscuring him into oblivion—but also the men who wear it (they're not worth the cracking). If Hank had instead said knights wear silly clothes and are useless we wouldn't feel the bite to the same degree—knights and their armor might be dismissed with lighter language, but not mocked as well.
It isn't all fun and games, though, which makes sense since the ultimate goal of satire is to expose ills—underlying the jokes are serious concerns. And so while Hank easily handles every challenge that comes his way, at the end of the day Merlin defeats him:
"Ye were conquerors; ye are conquered! These others are perishing—you also. Ye shall all die in this place—every one—except him. He sleepeth now—and shall sleep thirteen centuries. I am Merlin!" (44.2)
Hank's efforts to change the world end in defeat and he's dropped back into the present to die alone knowing that he's failed. That colors the humor with real sadness, as if Twain knew that the human race ultimately can't be fixed.