Probably the most blatant one which hasn't been officially confirmed as a Canon pairing is Snake and Otacon - they've managed to talk extensively about the meaning of love, and make several remarks about how the other taught them how to love. Add to this lots of snarky banter, the natural chemistry between the two as a Spy Couple, and Otacon's famous "I know Snake better than anyone else" line (a recurring line through the series, which is used as a sexual innuendo every other time it is said). Even the voice actors pick up on it.
David Hayter, Snake's voice actor, confirmed that the relationship between Snake and Otacon was supposed to be "romantic," and confessed that they'd played much of their dialogue as love scenes, particularly the 'Bad' Ending of the remake of the first game. Male and female reactions to this were predictable.
The "romantic" nature of their relationship is also evident in the second stinger at the end of MGS4, when Snake decides to live the rest of his (short) life in peace and Otacon doesn't hesitate to go with him. In addition, in terms of the main Solid Snake storyline, not only is this the last conversation of the entire game and series (and also possibly the last canonical conversation between Snake and Otacon that the audience hears), Snake's last word is Otacon's name in response to Otacon promising to stick with Snake until the end.
In Metal Gear Solid 2, Otacon monologues about how he found a 'happy family' and 'learned that [he] could love'. Since his love interest in the previous game was hardly family, it's natural to assume he's talking about Snake, who he seems to be living with.
In Metal Gear Online, Otacon will only sing the victory hymn if Snake is whistling with him.
One line: "You and your overactive libido." (From Otacon to Snake in Metal Gear Solid 2.)
It even shows up a tiny bit in Super Smash Brothers Brawl, during the codec call easter eggs. Otacon acts very irritated and jealous during the Samus call ("Yeah, well, Samus is deadly"), and in most of their conversations there's this flirty undertone, especially in the ROB call ("You sure know your geeky tech stuff, Otacon." "Heh heh, well, y'know..."). However, this may be due to the voice actors playing them as a couple as per aforementioned tradition rather than any intention of the writers. Still, considering that the codec easter eggs fit the tone of the MGS series basically perfectly, it's hard to imagine this wasn't intended.
In Metal Gear Solid 4, calling Otacon while hiding in a dumpster causes Snake to tease Otacon about how gross it is. They laugh together, and Otacon then says, in a flirty tone, that Snake needs to clean himself off before he's allowed to come back. Snake responds, in an equally flirty tone, "I'll see what I can do."
Not to mention that this is a Call Back to the conversation between Rose and Raiden if Raiden gets pissed on in Metal Gear Solid 2.
In Metal Gear Solid, there's a weird moment where Snake takes Otacon's shoulder and asks him if he's feeling okay, checking for symptoms of FOX-DIE. Otacon accuses Snake of 'getting friendly all of a sudden'. All that is pretty innocent. What isn't particularly innocent is that when Snake is accused of coming onto Otacon, he immediately jumps back a couple of metres and begins to act really coyly - "N-No, I'm just glad you're okay.". In The Twin Snakes, Snake does some much more invasive groping and it's shot through Otacon's eyes, and Snake looks really into it.
Not to mention Big Boss and Revolver Ocelot, which was eventually confirmed ascanon by the director in a Japanese commentary track.
To be more specific, Ocelot was quite in love with Big Boss. Big Boss saw Ocelot as more of an apprentice than anything, if even that.
Indeed. It was already very suspicious when Revolver Ocelot was shown to get turned on by torture for the first time because it was Big Boss getting it. EVA also hassles Big Boss repeatedly through the game about how infatuated Ocelot is with him; Big Boss remains oblivious to Ocelot's crush (despite the fact it can practically be seen from space), which results in EVA laughing at him. Even Gene from Metal Gear Solid: Portable Ops remarks on how Big Boss "caught Ocelot's eye," prompting Big Boss to ask confusedly what Ocelot has to do with anything. Ocelot disgustedly denies suggested attraction to EVA, yet fails to do so when the same is suggested about his obsessive crush on Big Boss. The commentary track confirming it was practically stating the obvious.
The statement given was that Big Boss and Revolver Ocelot shared a "pure love".
Revolver Ocelot often tosses double entendres in the direction of Solid Snake in the first game (Snake even flirts back during the torture room scene), and kisses him (on the cheek, though) in the fourth. Ocelot even moans 'Snake...' beforehand. It helps that, as noted above, Ocelot had a canon crush on Snake's identical clone "father", Big Boss.
When Snake gets kissed by Ocelot in MGS4, his Psyche restores. Yeah, Snake's.
And in Metal Gear Solid, during the first fight with him, he mentions how he loves reloading during battle. "'There's nothing like sliding a long silver bullet into a well-greased chamber." I mean, COME ON!
So does that make Volgin's "I know the Major better than anyone else" thing... still more proof? Oh, Raikov. You exist only to be used in this Crapsack World of ours.
Speaking of Raikov and Volgin, the MGS4 downloadable plot and character map on the PS Store confirms that Volgin and Raikov are lovers.
Really? I didn't find anything out of the ordinary about identifying someone by grabbing their dick and feeling its size. What heterosexual man hasn't done that?
Between Volgin, Raikov and Ocelot, one may begin to wonder if all Russian Military Officers are gay.
On the subject of Raiden (sort of), his relationship with Snake has to be of the Ocelot/Boss mould. Quinton Flynn does a great job playing Raiden's first encounter with Snake like a mixed-up Fangirl, both mad at Snake for using him like a pawn and jawdroppingly impressed by the fact that it was Solid Snake.
Forget Raiden; Vamp was sleeping with Scott Dolph, the Commandant of the Marine Corps, circa MGS2.
Pretty much every single line spoken between Big Boss and Python is Foe Yay-tastic, down to Big Boss recalling their 'training sessions' and Python yelling 'Feels good to me, how about you!' before returning fire during the boss battle. It's ridiculous.
Gray Fox did it first however in the first Metal Gear Solid. With lines like "Yes, hurt me more! More!" and having screaming orgasms every time Snake hits him, there's really no other way to interpret it.
Try taking a photograph of the anonymous manchest on the door of the open locker near the Engine Room in MGS2, show it to Otacon, and see what happens! Or, if that's not your thing, once you get into the third room of the Tanker hold, show Otacon a full film of photographs of Scott Dolph.
Otacon: Snake, if you like him so much, I'll print this out and make a panel out of it. Put it over your bed or something.
Also, you'll hear Snake say "good" if you manage to take clear enough shots.
EVA seems to have a sweet little crush on The Boss in Metal Gear Solid 3. She talks obsessively about how she's jealous of the very pure love between Snake and The Boss, since she herself only understands sexual love. And also moans The Boss's name in her sleep.
She also moans for "Snake" and "Fido".
Peace Walker has the rather aptly-named Strangelove ranting about The Boss, referring to her as "the woman I loved."
She'll also outright state that "naked men are not my cup of tea" if you call her while wearing Swim Trunks.
Big Boss and Kaz in Peace Walker seem to exist mainly as Hideo Kojima's attempt to outgay his previous effort of Snake and Otacon, in one single game. It's either the most deliberately homoerotic bromance ever, or they are genuinely in love:
Paz mentions in her audio diary that she saw Big Boss and Kaz fighting outside, completely naked, after Big Boss found out that Kaz was two-timing one of his MSF girlfriends. She goes into graphic description about the 'sweat flying off their muscular bodies', and realises at the end that there's no way she can get between their closeness. The whole thing reads like a slash fic.
It doesn't help that Big Boss beats Kaz up in the showers.
Attempting to go on a mission with no clothes on will make Kaz threaten to take off his own clothes 'if that's what you want'. And then he'll actually do it.
Pictured above, the mission "Date with Kaz". Your objective is to flirt with Kaz and call him to your love box.
In both versions of the game, Kaz taunts Big Boss about parachuting into the jungle 'completely naked', pretending not to understand what Big Boss really meant (he was unarmed). In the Japanese version, he follows this up with trying to persuade Big Boss to do naked skydiving with him sometime. Big Boss isn't too enthusiastic, though.
Voice actors ship them! The Japanese promo "Kazradio" podcast contains little, mostly improvised skits performed by the voice actors for Big Boss and Kaz. In one of them, they visit an onsen, and Kaz slaps Big Boss on the ass. Twice.
There's also this exchange:
Big Boss: Kaz, I want you to go behind me... there.
In Peace Walker, some of the missions involve looking through the doors of houses to find people inside. One of the 'incorrect' houses is surrounded by the sound of a shower. If you make Big Boss look through, he makes an aroused 'Oooh' sound, which is rudely responded to by a male "Who's there?".
Peace Walker is like one long MGS slash fic. Every other word is dripping with innuendo. Let's not forget when Cecile is telling Big Boss about her private baths from Strangelove, and when he seems bothered by the idea of Strangelove giving another woman a bath, she responds with a very sensual insinuation that she likes woman-on-woman as much as a heterosexual relationship.
And the homoeroticism continues in Rising! As Raiden, you can cut off the clothes of (male) civilians. Doing so will have your support team berating you for "non-consensual activities". Raiden will feebly defend himself by saying it was impulse on his part.
A rather infamous example from the first Solid game:
Snake: (To Miller) There's no one else I'd rather have in a foxhole than you.
Ground Zeroes piles subtext about Kaz and Big Boss. Paz calls Kaz's relationship with Big Boss "infatuation". Taken Up to Eleven in one of the Japanese-exclusive "Peace and Kazuira Blues" tapes, which details the "Gay Shower Fight" between Big Boss and Kaz. Words cannot do it justice, you have to listen to it yourself.