Axis Powers Hetalia didn't used to be as bad as some other fandoms about this, as shippers can find foundation for virtually any ship if they're good at Wiki-fu, and multishipping and Ship Mates are common practices in the fandom. Key words: used to be.
Of course, when you've got Loads And Loads OfAnthropomorphic Nations, a fair amount of Ship Tease, and a distinct lack of Official Couples, you can justify pretty much any pairing under the sun. The result is that people who ship Nation X with Nation Y will find themselves at odds with those who ship either X or Y with another character.
The most prominent example is the struggle between America/England fans and anti-fans. America/England is easily the most popular ship in fandom, but its massive popularity has caused many fans to attack it as "too popular" and "taking attention away from the much more interesting ships" (which sometimes means "gets more attention than MY ship", considering that Russia/America and France/England fans tend to be the loudest about this) It's reached the point where some America/England fans don't even post their fanfic/artwork at the main Hetalia community on LJ and many fans avoid the community as a whole because they feel that uneasy about the general attitude there.
You just know a fandom has a serious case of Ship-To-Ship Combat when the very first reaction to Portugal's prospect appearance is people using the character to attack America/England, claiming that Portugal is England's "true partner" and the pairing will "take down" America/England. Poor Portugal only exists as experimental sketches and doesn't even have a personality yet, and people are already gearing up for a shipping war, much to the annoyance of those who just want to ship in peace.
There are even some explicitly anti-America/England fanfics out there that are essentially rants placed in the characters' mouths about how America/England deserves to be hated/bashed for daring to be too popular. Truly, America/England is theBase Breaker ship of APH fandom.
The anti-America/England fringe try to cash on the Incest Subtext betwen them, falsely claiming that their relationship would be Parental Incest. This is despite how England refers to himself as "this child's older brother" in the "Battle for America" strips, and America saying "I'm not your little brother" anymore in "Cleaning Out the Storage", thus making this relative squick lower as they do not consider themselves as adoptive father and child.
Suggestions that America/England might be canon have fueled someone to spread rumours about the pairing being actually canon. Eventually, many fans of the pairing used it to claim some sort of superiority over Russia/America and France/England shippers. Needless to say, this sort of caused mass indignation on Tumblr and similar media, and some fans who disliked the pairing actually dropped out of the fandom out of hate towards it.
Rabid Poland/Lithuania and Russia/Lithuania shippers engage in Die for Our Ship against the other's preferred Liet suitor regularly. The former camp cashing in on Russia's status as the local Psychopathic ManchildandMemetic Molester to turn him into a bastard and ignore his tragic backstory as well as his Despair Event Horizon; the latter camp exaggerates Poland'sinitial selfishness to Jerk Ass levels and accuse him of being too shallow and brainless for Lithuania, erasing Poland's Character Development. (In case you're wondering what happens to Belarus in all this, she's usually just ignored completely because of her not being a major character and the Belarus/Russia and Belarus/Lithuania camps being much smaller and less vocal than the other, more popular rival ships like Russia/America, Russia/China, etc. But yes, STSC between the latter group of ships and Russia/Lithuania has been spotted from time to time, with Russia/Liet fans either making Belarus a monster who would rape Liet to make him too "impure" for Russia, or blaming her for Russia's inability to make friends.)
Sometimes Russia/Lithuania can go head to head with America/Lithuania, over which partner would be the healthier or more historically important for Liet. And sometimes Russia/Lithuania winds up bashed because it's felt Lithuania is "getting in the way" of Russia/America.
In Japan where England/Japan is one of the most popular pairings, there was a huge conflict between rabid shippers (most of them were actually trolls) of Japan/England and America/England that involved trolling, personal attacks, deleting each other's videos in Nico Nico Douga under false accusations, etc. Note, though, that it was essentially a 2ch shipwar getting out of hand and the majority of shippers (the sane ones) on both sides were dismayed by it. (In fact, America/England shippers on both fandoms generally either regard Japan as a best friend to both America and England or having one-sided feelings for England that he's not planning to act on; and England/Japan fans either ignore America altogether or use England's America-related angst as great opportunity for some hurt/comfort). A few America/Japan shippers got mixed up in that mess too, although in smaller numbers (America/Japan fans tend to portray England as the sympathetic unlucky rival and/or friend). Funnily enough, soon after the combat died out, there was a boom of fanart and doujinshi for the Outcasts OT 3 (America/Japan/England). Coincidence? I think not.
Let's not forget the rabid Prussia/Austria fans who insist that Prussia and Austria are obviously passionately in love with each other in canon and that Hungary is a second thought to them at best and an annoying obstacle / woman at worst.
Fans are divided on whether Greece should be with his rival Turkey or his friend Japan, with the respective viewpoints boiling down to "Greece and Turkey obviously have Belligerent Sexual Tension in canon and they make for a far hotter and more interesting ship than Japan ever could with Greece" vs. "Greece and Turkey obviously hate each other's guts in canon and Japan is the perfect match for Greece." Though it's really just the Vocal Minority of Turkey/Greece shippers who look down on Greece/Japan as "boring and non-historical" and Greece/Japan shippers who look down on Turkey/Greece as "abusive and non-canonical" who can get a bit snarky at times.
The worst by a mile, however, are the militant Turkey/Greece fans who attack Japan as a Relationship Sue with little to no evidence, the anime as promoting a "historically unimportant" pairing over a "historically important" one (but never the webcomic or manga, funnily enough), and/or the Greece/Japan ship as "unfairly popular" (even though Greece/Japan, while somewhat more popular than Turkey/Greece, gets nowhere near the popularity of the F/A/C/E ships, which makes the "gets more attention than my ship" sentiment behind this claim even more obvious). The majority of Turkey/Greece fans are not like this, of course, with many of them even shipping both Turkey/Greece and Greece/Japan, and the Vocal Minority is at least merciful enough to leave Japan alone in their fanworks, but that doesn't make the minority's accusations any less embarrassing.
Regarding Japan ships in general, there's some division between those who think Greece/Japan is a Moe Couplet and those who think it's just a snoozefest, those who think England/Japan is the true Moe Couplet and those who think it really isn't, and those who find China/Japan or Japan/Taiwan more interesting and "historically canon" than the aforementioned ships and those who find them squicky due to the conflicted history between them. Like with the Turkey/Greece vs. Greece/Japan debate mentioned above, it isn't too bad as multishipping is accepted. So long as you leave out the minority of Fan Dumb who like throwing around the terms "Crack Pairing" and "unfairly popular" in regards to the top two Japan pairings with little to no logic behind them, that is.
Japan/Taiwan seems to have a pretty bad reputation in the West. As it's the most popular pairing for Taiwan in J-Fandom and pretty much the only het choice for Japan that is mildly aknowledged, many W-fans side-eye it hard and extrapolate their issues with Taiwan's characterization in many artworks to their real opinion on Taiwan the character, accusing her of being a Relationship Sue and being harsh to everyone who ships it, saying that the Japan/Taiwan shippers are "creeps who ignore the issues of the Japanese colonization to imagine themselves as Taiwan" among other things.
Spain/Romano has also been under fire lately, as they're often accused of pushing their couple into other shippers' faces due to some Himaruya notes, or hating on America/England shippers for their pairing's popularity and accusing England of being a pedophile. Of course this isn't always the case and the latter accusation was actually exaggerated and used in propaganda by some anti-fans to discredit the pairing — but the Vocal Minority of jealous Spamano fans didn't help the pairing's case with their own pushiness and make the generalized accusations all too closer to reality to ruin it for everyone else.
Spain/Romano vs. Romano/Belgium has started to become this trope, due to a few fans on both sides having tendencies to tear down the rival pairing and invalidate it as "incestuous and sick". Anti-Shipping Goggles also play heavily into the debates of who Romano really likes. To some, a third option doesn't even exist because there's just no possibilityof him being bisexual — not to mention, it adds more fuel to the Het is Ew camp's unfair "Belgium and the other Hetalia girls are only stupid and anti-feminist Relationship Sues!" accusations. Throwing in Spain/Netherlands may or may not softened this, as it offers a chance to get Spain away from Romano without recurring to Die for Our Ship.
Spain/Romano vs. Spain/Belgium is another bit of a feud similar to the above. The Spamano diehards claim that Belgium is "in the way" of their pairing, claim that she's nothing but a Mary Sue, and direct immature hate at anyone shipping her with Spain. The diehard SpaBel fans claim that Belgium is more important in Spain's life than Romano, that he "canonically" would like her more since his alternate universe counterpart liked her, and that Spamano is promoting incest.
Some France/Spain fans do not take too kindly to Romano's abrasive personality and the idea that Spain would be more attached to a "historically insignificant" colony of his than a "healthier" relationship with France, who he was said to have an undesirable yet inseparable relationship with. Though it's still yet to reach a toxic level like with America/England's involvement in shipping combat, a vocal group of fans have thrown potshots at Romano and accuse him of being in the way of Spain's happiness with France or any male that's not "an abusive, heartless psychopath" that Spain is too dumb to see the bad in. Oh, and don't forget the Romano bashers who insist on Romano being a bigot and an homophobe in canon too.
Prussia/Romano has become a recent rival for Spain/Romano, with certain fans from the Prumano camp perpetuating claims that Spain's a pedophile and too idiotic to truly love Romano and that Prussia would be a kinder and more supportive lover. Those against Prumano believe that Prussia and Romano would be too dysfunctional for each other, based off of Romano's dislike-to-absolute LOATHING of Germanic nations (although Spamano isn't exactly a dysfunction-free pairing in itself).
The biggest victim of these ship wars, ATM? Seychelles, the local Nubile Savage and Plucky Girl. She was a minor character with a mere cameo in the comics... then she became an Ascended Extra in Gakuen Hetalia (both the original script and the PSP game), and then the poor girl started getting shit from everyone coming from everywhere. The Het is Ew camp called her a Relationship Sue, the "fandom feminists" accused her of being "offensive" and "submissive to males", and the USUK and FRUK rabidshippers accused her of standing in between their beaus because she had some degree of Ship Tease with both England and France. Nevermind that France is after Anything That Moves, that Gakuen Hetalia isn't that centered in romance but more in The Power of Friendship, that Seychelles isn't exactly shown as submissive or an Extreme Doormat to either France or England, and that in the PSP game itself Sey actually helps America to regain England's attention and affections after a fight, which can easily be interpreted as her Playing Cyrano for them.
Prussia/Canada gets Fan Dumb from both sides, thanks to garnering both rabid fans who tout it as indisputably canon and interpret everything remotely related to either character as ironclad proof that Himaruya is shipping the two together and rabid haters who attack it as an insultingly baseless pairing that only shallow fangirls who don't give a whit about history or canon would think is remotely plausible. The first camp is definitely annoying considering that they've done things like declare that Prussia's character song was obviously a love song to Canada on absolutely no evidence or applying Die for Our Ship treatment to Cuba and France for no other reason than feeling "threatened" by their bonds with Canada, but so is the second camp who often acts as if the mere popularity of Prussia/Canada is an affront to the entire fandom in spite of it being far from the only Hetalia pairing that gets little support in canon or history.