In the Watcher's Keep area of Baldur's Gate II, there's a quest where the PC or someone in his/her party takes on the avatar of the Spirit Warrior, and that part of the game plays like way old-school Rogue-like RPG's, where the player's moves are controlled by menu options after reading descriptions of what the avatar sees in the room of each dungeon.

I'm also thinking of this game within Space Quest, where Roger is trying to kill a chicken and very silly music is being played. My memory is completely fuzzy on this, but I know it's there b/c it was the only thing I was good at when I used to play SQ as a youngin.

Does Super Turbo Turkey Puncher count?

Comic Jumper allows you to play other Twisted Pixel games from your home base, they are accessed via some stand up arcade cabinets.

Magicka is a game based around easter eggs, even the chapters are named after star wars films.

Kalisto's Fury of the furries is swarmed with homages and references. From the title cards on the worlds (jaws, indiana jones etc), to background objects (romeo and juliet scene playing out, crashed walker from Star wars episode VI in the jungle levels) to enemies (T-1000, why won't you following me trying to kill me?) to entire levels (There is in the later part an entire level setup like the pac-man grid, with ghosts, and you're still playing a platformer, also I'm fairly certain there was a pinball level, again you're playing a platform game).

Also if we're talking about games within games, Prince of Persia: Sands of time features an unlockable full version of Prince of Persia 1 if I recall.

Pitfall (the PS2 game) has secret areas where you can play the classic Pitfall 1 and 2 from the old Atari system.

System shock 2's final area is in large part a recreation of parts of the first floor of System shock 1

Serious Sam (forget if it's first or second encounter) has Sam enter a phonebooth, calling up and asking for Blondie, asking if he'll be here soon, seems like he's been waiting forever...Well forever finally happened.

Ha, I never heard of that one before. I'll try and find it on youtube.

Chicken Invaders 4 has an world dedicated to Space Invaders

World of Warcraft may not play like earlier games but in game it pays homage to literally tons of both earlier games and cultural references. I can't find a central repository for these but the most recent that I've found are the jewelry dailies in Stormwind. I believe all four make reference to Futurama.

Edit: fixed typos

Oh, non-level tributes count to?
Well, i'm not really into Star Wars, so i don't know if this place was in any SW game.
But Crusader: NR/NR features a Resistance outpost called Echo Sector, which is kinda like the Rebel Echo Base in Empire Strikes Back. The main character being a red Fett is also a sign, though, many people saw this as a ripoff, not homage.
Crusader also references System Shock, implying that a new AI is being installed on the new Citadel station, my manual is in Polish, and the GOG one is some cut down version, so i can't give You the exact quote.

Not a level, but still fun:
In one of the Monkey Island games Guybrush Threepwood (Mighty Pirate tm) comes across a skeleton wearing an "Ask me a about Grim Fandango"-badge, although this is probably more advertising than an Homage.

Space Quest 4 is all about traveling through time to other versions of Space Quest.

Honourable mention: in Crusader: No Regret there is a shout-out to Ultima 8; at one point you can find a secret area with a pentagram used to summon Pyros from Pagan.

Space Quest 1 SCI has a King's Quest 1 screen. When You use the escape pod the wrong way, You get warped to KQ1, and crash somewhere there.
SQ4 also features Leasure Suit Larry 4 [which was never released] as a deadly computer virus that helped obliterate Xenon [as far as i remember it anyway].

Duke Nukem 3D does Quake: "I ain't 'fraid of no Quake." [L.A. Rumble level]
Duke 3D does Doom: "Another doomed space marine" [Death Row level]
Blood does Duke: "Shake it Baby" [Dark Carnival level]

I just saw that on youtube this morning... hilarious.

And in the original release it was "Ask me about LOOM" if your reference is the badg on the pirate in Monkey Island 1 near the beginning.

And Monkey Island 2 had Sam and Max costumes in the costume shop.