From this gallery page, A New Life Era and Moe and BIOS birthday party heavily speculate this between between Amy Jr. and Moe. The Recovery Part 2 shows there MIGHT be some romantic love between the two. However, they're both very young, and may not be each other's soul mates. For all we know, Bertha might be Moe's soul mate, although The Hacker probably wouldn't be too happy about that.
While the series itself doesn't go much beyond some swooning and flirting as far as the Rangers themselves are concerned (except for Monterey Jack who used to be engaged to another mouse), Fan Fiction does, and a lot, even those avoiding flat-out Shipping fics. It usually builds on ships already hinted at in the show. The most common Interspecies Romance is between Dale (chipmunk) and Foxglove (bat).
Also quite frequent is the "Pro" one between Chip (chipmunk) and Gadget (mouse). "Pro" fanfics are more common than "Anti" ones which ship Gadget with someone else or have her prefer celibacy to a relationship with Chip.
While Dale and Foxglove can be seen as a couple at the end of "Good Times, Bat Times" already, Chip and Gadget usually don't start out in a relationship at the beginning of a fanfic or a new Fan Verse. Fics and even 'verses like the idea of letting the actual romance grow by itself (for example Fly to the Light) or leaving one or both heartbroken because they Cannot Spit It Out and putting one or both through the wringer before their deserved First Kiss (for example Rhyme And Reason, Of Mice And Mayhem, half the Midnightverse).
Of Mice And Mayhem is a special case. Chip and Gadget are strongly in love with each other, and they do come closer, but only as close friends because neither wants to take the next step due to genetical incompatibility. Especially Chip thinks that Gadget might want to have children one day, and that she'd better wait for a mouse guy. Later in the story, while Chip is still in Washington and the national mascot, Gadget performs a DNA check on herself on the background of the events of "A Fly in the Ointment" when she switched bodies but not heads with Dale, and she discovers that there is still enough chipmunk DNA left in her to make her compatible with chipmunks. Dale and Foxglove, on the other hand, simply don't care, or they don't want kids.
Next to Dale & Foxglove and Chip & Gadget, The Midnightverse hints at something growing between Sparky (rat) and Lahwhinie/LaWahini (mouse). Diamonds In The Desert reintroduces Clarice from the Chip N Dale cartoon "Two Chips and a Miss" who plays with the two 'munks, but who also is the crush of a mouse. It also introduces a third rat who looks a lot like Rat Capone and Francis (they're brothers in this 'verse), and who hits on Gadget a lot.
It is said that The Untold Ranger Tales haven't left out a single possible straight pairing among their characters yet, and they're as multi-species as CDRR fanfics come.
In A Charmed Life Light determines he needs to be closer to Ryuk to avoid the Shinigami getting bored and killing him. He becomes closer to Ryuk than anticipated when he discovers Ryuk has a crush on him.
Zim and Dib (Irken/human) are by an enormous margin the most popular pairing in fanfiction, due to perceived Foe Yay. Less popular variations include Dib and Tak (human/Irken), and Gaz and GIR (human/robot, though granted he did show signs of having a crush on her).
Twi and Me has human male and female pony (Twilight Sparkle).
Xenophilia involves Rainbow Dash getting a human boyfriend. Despite being a Lemon the fic spends a fair amount of time discussing differences between humans and ponies as come to sex and gender relations. It mostly treats the human as the alien partner (talking about his Exotic Equipment, for instance).
The Mixed Up Life Of Brad has this between Brad the human, and Twilight the unicorn. Subverted a bit, as Brad fell in love with Twilight when she was transformed into a human, and now that she's back in her pony form he finds her physical appearance unappealing. It is noted that they cannot ever be truly happy together, being so different from each other.
The same author's A Mighty Demon Slayer Grooms Some Ponies gives Danny (from G1 My Little Pony), an eleven-year-old human, a crush on Surprise the pegasus (who is an adult at the time by pony standards). Nothing comes out of it, and he still has bad memories years later. It's hinted that it's all because he was still a young kid at the time, confused by his budding sexuality.
In Arads Stardust, it's fairly common in Equestria, in part because there are an average of 3 stallions to every 7 mares. Upon hearing this, Lana immediately figures the best way get Twilight off the fact that she's XCOM's newest test subject is to imply that Matt thinks Twilight is cute. She takes it very seriously, and even searches for information about "Human Courting Protocols." Matt is mostly upset how Lana refuses to stop teasing him about it, but there is some Ship Tease between the two.
Mating Customs of Species Equus parvus explores this a bit: Humans have a tendency for seeking exotic partners, an evolutionary trait which used to help them reduce inbreeding by encouraging them to find partners in other communities. As a side-effect, it makes them prone to Interspecies Romance with other sapient races—which, however, do not share this trait, so such infatuations are pretty much always one-sided (and result in the unlucky human suitor laughed off).
Arddun Lleuad and its sequel, Chwe Goleadau, has the romance between the human protagonist and Luna as one of its main themes. They have some anguish over just how far are they going to take their relationship. At the very end, the point becomes moot, as the protagonist gets castrated by the villain.
In All-American Girl, there are close relations between Earth and Equestria, and pony-human relationships or couples are not unheard of—the protagonist herself is a pony married to a human. That said, the setting also has temporary species-transformation magic available for the spouses, allowing such couples to actually have children together.
In Summer Days And Evening Flames, Gilda (a griffin) eventually starts dating her guard captain Iron Bulwark (a pony). She's justifiably worried that dating him could make others look down on him, considering his best friend is very racist to griffins.
Nearly any fanfic where Gilda, Spike or Discord are paired up with anypony, as they're the only ones of their species with any prominence (though Discord is closer to Did You Just Romance Cthulhu?).
A pair of recurring background characters in Cadence In A Minor are a stallion/griffon couple, who are also soldiers on the opposite side of a tense border. Despite everything stacked against them, they make a pretty cute couple, all told.
The Reading Rainbowverse has the typical Gilda/pony romance... but mixes it up by having a cat crush on Gilda too. Not an anthropomorphic cat... a normal housecat.
Strife, the Draconequi of Natural Selection, is Happily Married to Mortis, the Alicorn of Death, and have kids.
Dark World!Derpy has feelings for the Doctor, which is part of her motivation for wanting to destroy the Valeyard and force him to regenerate back into the Doctor. She succeeds and ends up marrying him after Nightmare Eclipse's defeat, though he's using a Fob Watch.
Celestia and Discord were once in love, but Discord's Sociopathy and sadism ended that.
Minuette and Captive Audience, as Minuette is actually a Time Lord. Though it's a bit ambiguous after Minuette destroys the Master if she's still one or not.
TD ends up in one with Cheerilee in the main continuity of The Non-Bronyverse. In the alternative-continuity AU story where he gets accidentally turned into an alicorn, he eventually strikes one up with Celestia.
Another Gardevoir related fanfic. This one is about a boy growing up with a female ralt. Due to some genetics, the ralt, then kirlia comes more and more human as she spend time with the boys and as their relationships develops.
Poor Ash has been paired with practically every character under the sun, human or Pokemon.
Subverted here. It opens with a great big monologue about human and pokemon being intimate — even having offspring — is supposedly very common. The titular character has had two inter-species relationships: One with Tera, a pelipper, the other with Henry, a human. Neither of which was intimate. Tera just wants to be a dad, while she Henry just gets a kick out of tricking people into thinking they're a couple.
Jack Frost, originally a human, is very often times paired with Bunnymund, a Pooka (a large, sentient rabbit species, possibly alien in origin), by "Jackrabbit" shippers in fan fic. Indeed most fan fics that include a romance with Bunnymund are this.
The Kirk/Spock pairing from many a fanfictionnote also the Trope Namer for Slash Fic. Kirk is human, Spock is half-human, half-Vulcan.
The fic Written in the Stars deserves mention in that Kirk has been genderswapped, and ends up with Spock in four different realities. Each time resulting in a Half-Human Hybrid daughter.
The pairing of Captain Picard and Q is rather popular in fanfiction. Picard is a human, and Q is a transcendent energy being, who can shape shift at will and warp reality like it was a toy. Fans are fond of exploring not only the various differences between Q's god-like perceptions and Picard's idealistic mortal worldview, but the inherent power struggle that comes from Q's powers and Picard having to use his mind and challenge himself to try harder to stand toe to toe with Q's antics. As a hyper-intelligent and virtually omnipotent being few things manage to excite Q, so Q's fascination with Picard as a particularly special human is given a lot of focus.
Bait and Switch has one between a couple of Mauve Shirts from the USS Bajor's security department, Ensign Kate McMillan (human female) and Lieutenant (JG) K'lak (Klingon male). They're stated to have been in an on-again, off-again relationship for the last couple of months, but decide to go exclusive in chapter nine.
In Red Fire, Red Planet Lieutenant Commander Brokosh, a Lethean mercenary, is Happily Married to Klingon noblewoman Ba'woV, daughter of N'Gara, of the House of Chel'toK, a Klingon. Brokosh signed with the Klingon Defense Force when their son was conceived, since Chel'toK is fairly impoverished.
In one 4chan copypasta fic King Arthur in a Peggy Sue scenario decides that he's better off in making that squirrel (who actually loved him and he has Merlin change into a human form) his Queen instead of Guinevere.
Although the fanbase for this pairing dates back somewhat to the 4Kids series, Donatello and April O'Neil has had a wave of momentum because of Don's crush on her in the Nickelodeon series, spawning numerous fanfics and fanart, Another popular couple inspired by both 4Kids and Nick eras are Leonardo and Karai.
Fans of the 1987 animated series have Donatello/Irma, April/Michelangelo, Michelangelo/Kala, and Leonardo/Lotus Blossom.
Time Will Tell has a romance between Jorryn (the woman from America) and one of the hobbits, but it takes a long time to develop.
Some authors use a Metamorphosis, by which the daughter of Men suddenly becomes a elf or a hobbit. This is a big hint that her love interest will be another elf or hobbit, and a same-species romance will ensue.
Many fan authors for The Hobbit have no concept of a female dwarf. Thus dwarves like Thorin, Fili and Kili are always falling in love with some elf or mortal woman. Dwarf with dwarf romances are rare. Worlds Apart has a dwarf with elf romance.
The Sirens of Azeroth suggests that the many species of Azeroth don't have a huge amount of trouble with, at least, casual interspecies romance/intercourse, demonstrating among others a Dwarf/Draenei (seven-foot faun-like blue alien) relationship, a Troll/Blood elf lesbian couple with an open relationship, and a pair of tiny gnomes talking about how their favorite types of 'men' are Tauren (hugeminotaurs), though they could just be making that up.