Konami makes multiple references to their past games, the most memorable being in Metal Gear Solid, where Psycho Mantis will comment on your love of Konami (or Nintendo games, if playing on the GameCube) games by name as a show of his powers before the big fight with him. He's reading your memory card: for more details, a guide about the Konami games detection mechanics for the Japanese PS1 version of Metal Gear Solidcan be found here.
Statues of giant mecha from one of Kojima's other games adorn a table in Metal Gear Solid 3, and in MGS1 a Policenauts poster is on Otacon's lab wall. In the Nintendo GameCube version of that game, there are action figures of Mario and Yoshi.
Metal Gear Solid features the character Hal Emmerich, who goes by the name Otacon — and his official bio makes it clear that he's chosen that nickname in honor of "Otakon", the US-based anime and east-asian culture convention (the name is used with the convention's permission). Dr Emmerich is usually credited in the convention's program guide as their "Scientific Advisor".
Kojima seems to have a love of 80s action movies
The cover for the first Metal Gear was based on a promotional still of Kyle Reese (Michael Biehn) in The Terminator
The faces used for characters in Metal Gear 2: Solid Snake are based on action stars of the time. In particular, Mel Gibson is Solid Snake and Colonel Campbell is Richard Crenna (Col. Sam Trautman from Rambo). These were replaced in subsequent re-releases.
Solid Snake uses the alias Iroquois Pliskin, a direct reference as to his namesake and further point to Hideo Kojima's fondness for Escape from New York and "Snake" Plissken. It's well known that Solid Snake and Big Boss were designed off of Snake Plissken.
Let's not even start on the Shout-Out theme naming of the characters. Just taking the above example, 'Hal' is from 2001: A Space Odyssey (as is Snake's first name, David), and 'Emmerich' is a reference to Roland Emmerich, a director who Hideo Kojima likes.
The 2001 reference was lampshaded in the Otacon end of MGS 1. They tell each other their real names, and notice the coincidence. "Hal and Dave, that's a good one. Let's hitch a ride to Jupiter."
Scott Dolph was the foreign co-ordinator for Metal Gear Solid, making a cameo voice-acting a random soldier, and being responsible for reciting all the dialogue on the Metal Gear Solid 2 industry inside trailers ('It has been two years since the Shadow Moses island incident...'). The sequel has a character named Scott Dolph who "likes making speeches".
Interestingly, Solid Snake worked for FOXHOUND early in the series. Snake Plissken was played by Kurt Russell, who actually voiced one of the title characters in a Disney movie titled The Fox and the Hound. Coincidence?
It's also well aware of the Follow the Leader culture it started, and makes some pretty affectionate references to other stealth games. The most blatant was easily the one where they had Snake dress as Splinter Cell's Sam Fisher in a gag trailer, but there was also the fact that you could find dog tags belong to Jean-Luc Cougar, protagonist of Win Back; and a gag conversation in which Snake suggested that 'Sam or Gabe' could do his job just as well as he could. (His conversational partner breaks the fourth wall to tell him to cheer up - 'This is the genre we turned over every leaf in!')
A more recent gag trailer had Solid Snake dressing as Alta´r of Assassin's Creed I. Then Kojima confirmed that it would actually be a bonus outfit in Metal Gear Solid 4!
Metal Gear Solid 3 has several shout outs to the James Bond movie series: the Cold War theme, surreal opening sequence with accompanying jazzy song titled after the game, a Bond Girl (EVA), and of course, the post-mission sex scene. Gadgets don't count since they're available in all MGS games, although the Fake Death Pill and stun gun disguised as a lighter are so James Bond-like.
The first thing Snake says after he lands during Operation Snake Eater is 'Can you hear me, Major Tom?' which is a reference to the David Bowie song Space Oddity. Originally, the Shout-Out would have been more significant, as the whole game would have been about the Space Race and Kojima wanted to use the song (which foreshadows Zero's role in 4) for part of the credits.
Metal Gear Solid 4 also has a few references to Star Wars, Darth Vader in particular. In Shadow Moses, right before you ride the elevator up to REX's hangar, you can see Darth Vader's helmet sitting right on the lift. Also, when Snake is sitting in the USS Missouri, as everyone is talking about Outer Haven, Snake says "His very own Death Star.
Back in the 90's when Konami was known very well as the King Of The Shoot Em Ups, they made Parodius-series which tributed to Konami's best in the 90's, and not only that, to the famous shootem ups back in that time, due the expy weapon sets ripped straight from Konami's shootem ups and even 3rd party shootem ups like Irem's R-Type, Taito's Darius, Toaplan's Truxton-series, Seibu Kaihatsu's Raiden, and so on.
Jikkyo Oshaberi Parodius, particularly its Forever With Me-port for PlayStation and Saturn, is full of Mythology Gags and references to other Konami games, like Ganbare Goemon, Twinbee, Lethal Enforcers and Tokimeki Memorial. Sadly as it's Japan-only, many miss the references and parodying jokes poking fun at the games. One guy seemingly tries to cover them up with his own response to History Behind Smash Bros with History Behind Parodius.
Astute Konamiophiles will note, though, that the Yu-Gi-Oh! card game is the king of this trope, with cards designed after characters from games like Gradius, Castlevania, Goemon, Metal Gear, and even Sexy Parodius. Also, a strong rumor among the TCG fandom has the translated name for the card Sky Dragon - Osiris (Slifer the Sky Dragon) as a Shout-Out to 4Kids employee Roger Slifer. Added to this in a recent Deck release was the card that translates to "Different Dimension Revival" however, the card in Japanese has the English letters DDR as the primary name for it, a direct reference to Konami's dancing game Dance Dance Revolution.
Building on that last comment, in the game DDR:X, there actually is a song referencing the Konami code. It starts out with someone singing "Up, up, down, down, left, right, left, right...SELECT START!" Funnily enough, it's a love song.
pop'n music takes this a step further, with Vic Viper, Lord British, and SimonBelmontBelmondo as playable characters. Vic in pop'n takes its NES Gradius form and even comes with the scrolling starfield and HUD from that game, Lord British isn't actually LB but merely a Palette Swap of the Vic Viper, and Simon is in his classic blonde form (as opposed to his currently-canon version, who sports red hair).
Really obscure in the DDR Hottest Party games on the Wii, since you have to be failing to hear it, but sometimes when you're in danger, the announcer will say "Dancer needs groove badly".
Snatcher contains a sequence in which the main character sneaks into a bar called Outer Heaven. It's the masquerade night, and everyone is dressed as a character from a previous Konami game - Goemon, Simon Belmont and Dracula, Bill and Lance, and Mr. Wu from Lethal Enforcers, to name just a handful. The main character can even dress up as an Easter Island head, which show up a lot in Konami games as a shout out to an employee who apparently greatly resembled one. It's hand waved as a current fashion for worshiping old pop culture, and is incredibly blatant when the main character starts complaining to his Robot Buddy about how he hated not being able to jump on the stairs in Castlevania, to which the robot responds, "The suicide rate amongst teenagers tripled that year." Let's not forget that the Robot Buddy is named Metal Gear...
And, in a shout out to a shout out, the "Metal Gear Mk. II" from Metal Gear Solid 4 is virtually identical in appearance and function to the Robot Buddy from Snatcher, except that it's not actually a character in its own right — Otacon provides its "lines".
The Easter Island head is very common recurring joke in the Castlevania series, though.
Also from Dawn of Sorrow and Portrait of Ruin hails the boss Zephyr, a humanoid creature that can stop time and shouts "Toki wo tomare!" when doing so. He also throws knives when time has stopped. Sound familiar?
Jonathan and Charlotte in Portrait of Ruin can perform a combination spell called the "Axe Bomber", which has them pincer-attacking with a dual crooked arm lariat, complete with blue lines of "magnet power." Cross Bomber much? Jonathan can also assume a defensive position that allegedly has been handed down from generation to generation in his family - much like the Kinniku dynasty's similar-looking Muscle Curtain.
And don't forget the boss Goliath in Order of Ecclesia. A hulking figure who has an attack where he takes big steps, and if he corners you, he gives you a series of rapid punches that will kill you, and when he dies, he lifts one of his arms upwards and a light pillar engulfs him. Yes, that's Raoh.
Not to mention one of Shanoa's glyphs lets her throw gusts of wind with an upward-sweeping arm motion, throwing 2 of them in rapid succession combined them together into a much larger gust, and the special attack for the glyph involves a cage of energy enveloping her. GeeseHoward anyone?