In the case of HIAWATHA, I'd go so far as to say it's almost impossible to take seriously now -- when one reads of his heroic warrior putting on his mittens before venturing forth, the effect is certainly not what Longfellow intended (they're magic mittens, but still -- thank God Gygax & co. called their versions of this gauntlets of ogre power instead). And it turns out Longfellow's Indians really do say "Ugh" in what he considered a suitable line of dialogue. It's interesting, therefore, to learn that it was already being parodied a short time after it came out,** or to find out that one of Longfellow's friends comforted him by saying it would long outlive all parodies and indeed the Native Americans themselves. Luckily the latter statement turned out to be as wholly untrue as the former; indeed, I'd say anyone's more likely to encounter a parody than the real thing, and that this has been the case for a long time (as Looney Tunes's Little Hiawatha shows).
It's also interesting to learn that accusations of plagiarism over too-heavy borrowing from THE KALEVALA also first popped up soon (two weeks!) after publication. Longfellow himself seems to have claimed he took only the metre from the Finnish and none of the characters or incidents for his story, which is self-evidently untrue to anyone who reads both works. It's ironic that Poe, who'd repeatedly and rather unfairly accused Longfellow of plagiarism in the past for what he felt were too-close echoes of Tennyson, had died six years before HIAWATHA came out -- a bit of a lucky break for Mr. Longfellow, one suspects.
All that aside, my Tolkienist's eye caught an interesting echo of JRRT's work in the final lines of Longfellow's poem, when Hiawatha departs in his canoe and sails off into the sky to join his father, the West Wind (emphasis mine):
On the shore stood Hiawatha
Turned and waved his hand at parting . . .
Launched his birch canoe for sailing . . .
Shoved it forth into the water
Whispered to it: Westward! westward!
And with speed it darted forward
And the evening sky descending
Set the clouds on fire with redness . . .
Left upon the level water
One long track and trail of splendour
Down whose stream, as down a river,
Westward, westward Hiawatha
Sailed into the fiery sunset
Sailed into the purple vapours
Sailed into the dusk of evening
And the people from the margin
Watched him floating, rising, sinking
Till the birch canoe seemed lifted
High into that sea of splendour
Till it sank into the vapours
Like the new moon slowly, slowly
Sinking in the purple distance
And they said: Farewell for ever! . . .
Thus departed Hiawatha . . .
In the glory of the sunset
In the purple mists of evening . . .
To the Islands of the Blessed
To the kingdom of Ponemah
To the land of the Hereafter
--Now, to me that sounds v. like a description of someone taking Tolkien's Lost Road, the old straight path to Eldamar and Tol Eressea and Valinor. And, interestingly enough, this parallel isn't nearly as strong in THE KALEVALA (or at least in the two translations I have of it, by Kirby and Magoun). It might be worthwhile checking out the cognate passage in the Old Kalevala.
Perhaps just coincidence, but Tolkien was at least aware of Longfellow and his work, since in one interview when asked if he'd like to be remembered as a writer or a scholar he compares himself to Longfellow, saying people remember Longfellow for Hiawatha and one or two other things and forget he was a professor of modern languages.
*although I was already familiar with the story from having read (and re-read and re-re-read) the CLASSICS ILLUSTRATED version, wh. turns out (as was so often the case) to have been extremely faithful to the original. I think the only work I learned from CLASSICS ILLUSTRATED that I've still never read in its original form is James Fenimore Cooper's THE SPY -- but then, having had to read one Cooper novel, I've never been able to force myself to do it again.
**the all-time best such parody being, of course, Lewis Carroll's HIAWATHA'S PHOTOGRAPHING, which uses the metre with far more dexterity than Longfellow ever does.