I'm going with Majora's Mask, without a doubt. Let's go through the games real quick...
MM - Let's start at the beginning of the game. Link is riding on Epona, everything is all fine and dandy, until Skull Kid comes in, steals your Ocarina and Epona, and takes off, Link clinging on to Epona's leg for dear life. You then follow Skull Kid into a sort of cave. You see him there, holding your Ocarina, mocking you. You are then turned into a Deku, and have to find your way out of the cave. You then meet the mask salesman, who wants his mask back from Skull Kid. Okay then, simple enough. Then we go out into Clock Town, and we've got three days until the end of the world. Whoopee. Three days to get the Ocarina and Mask from Skull Kid in your helpless Deku form. While later, we learn that we can use the song of time to go back to the first day, and that we aren't trapped in Deku form, the apocalyptic concept is still very dark.
TP - How does TP start? Rather Jolly, I would say. The game starts with you talking to Rusl, and then when you have control you're doing "chores" and stuff around Ordon Village. You do some goat herding, then you do some fishing in order to feed (Sera's?) cat, play around with little kids using a wooden sword, and the only real conflict that happens in the opening stages of the game is the part where Talo get's kidnapped by a monkey. That's nothing compared to what happens later in the game. MM immediately gives you a taste of what the game's dark plot is, wheras TP had a rather happy inroduction.
Introduction - MM > TP
MM - Because of the introduction, we basically already know what the plot of MM is. You have three days to defeat Skull Kid and recover Majora's Mask to return to the Happy Mask Salesman, while trying to stop the Moon from crashing into Termina. An apocalyptic plot like this is darker than many M rated games. While those M rated FPS, gory games have dark plots, about war or other situations that involve the need to shoot others heads off, few of those games have the subject of the apocalypse. In Majora's Mask, the world is going to end in three days. In those other games, the world is being taken over by the antagonist and his companions/minions/servants. I think it's safe to say that MM has the darkest plot among most games released so far.
TP - TP has a plot similar to most other Zelda Games. The antagonist, in this case, Ganon/dorf accompanied by Zant, are trying to take over the land, in this game it's Hyrule, like most games. This kind of plot is what you see in most games. Bad guys want to take over the world. Though this may seem a bit darker than normal, because rather than just a normal "take over the world", Ganondorf and Zant have a goal to cover Hyrule in darkness. I'm still going to say that MM is darker on this one. While it may seem pretty morbid for the Hyrule to be covered in darkness, MM is the apocalypse, the world ending. That's more extreme than TP, so I think MM can be labeled darker in this situation.
Plot - MM > TP
MM - MM has many events as you progress through the game that may not seem so cheerful all the time. Let's start with the first of the four main areas in that game, the Southern Swamp. The Deku people live in misery, as this area has been poisoned because of the disappearance of the guardian god. Link must help the suffering Deku's by defeating Odolwa and embracing one of the four giants. The next area is Snowfall. I found the atmosphere of this place pre-dungeon pretty miserable. Goron's cannot stand the cold. For them to be living in this freezing area seems pretty miserable. All the Goron's you meet here are in constant pain. In this area you also get the Goron Mask, which unlike the Deku mask, was made because of a great leader, Darunia, dying. There is so much death in this game compared to the other Zelda's. It's just something else. The next area is Great Bay. Right when you get here, you see, oh, guess what, a dying body in the ocean. You get the Zora's mask because of Mikau's death. And just like the previous two areas, the Zora's have quite a dilemma with their habitat. Great Bay is cursed and the ocean is in disorder. Once again, Link must go great distances to save the land. The last of the four areas is Ikana Canyon. Ikana canyon is known to be a cursed land, as it is home to the dead and filled with spirits who have much regret and hatred. After all four giants from the four area's are freed, you must go face the moon yourself, as it is about to crash into the earth. Overall, the events throughout MM are very morbid and dark compared to all other Zelda games, and all other games in general.
TP - As with the plot, TP's events throughout the game are similar to all other Zelda games. Most of the people throughout Hyrule seem to be rather happy in this game. They are going along, doing what they do in their everyday lives. Though some people in Hyrule are being affected by Zant and Ganondorf, and if Link doesn't stop them, everyone will be affected by the darkness Zant and Ganondorf plan to engulf the land with. I will say that TP has dark/morbid events throughout, but it doesn't compare to that of MM. There is just so much death and despair throughout that game, and that just seems to be much darker than any other Zelda game released to this day, and most video games released in general.
Mid-Game Events - MM > TP
MM - As far as an overall feeling in the graphics goes, MM is pretty neutral on that. The graphics aren't particularly dark throughout, as they are basically the same as OoT. Some things about the graphics are pretty dark, though. Some of the enemies are pretty dark graphic wise, like Dexihands, Wolfos, Desbreko's, and more. Though overall, like I said, the overall tone of darkness in MM's graphics is average and nothing to make a particularly big point about.
TP - The graphics in TP are probably the reasons there is an argument over which game is darker. The graphics in TP were pretty dark, to say the least. The main thing here is the Twili. Everything with the Twili, when you are in Wolf Link and have to go through those twilight sections, was pretty dark. A lot of those enemies are pretty odd looking and gruesome/dark for a Zelda game. Also, the overall realism in the graphics just makes the game's graphics dark. For example, the Redead Knights, the Puppets, and the Baba Serpents can seem pretty odd graphic-wise in a realistic kind of way, and can give the game more of a dark feel. So overall, graphics are one of the things that TP beats MM in when we are comparing the overall feel of darkness in these two games.
Graphics - TP > MM
Ending - Resolve to Initial Conflict SPOILERS
MM - At the end of MM, when you beat the game, Link successfully stops the Moon from crashing into Termina, and everyone is able to go back to their everyday lives without living in constant fear of the apocalypse. When Link returns Majora's Mask to the Happy Mask Salesman, he says the evil has left the mask, and he mysteriously disappears, bidding farewel to Link. Then, Tatl explains to Link that he should get back to his original quest, and Link and Tatl part ways. At the end, Link rides into the Forest once more. Endings in games aren't really ever generally dark most of the time. The conflict was resolved, as with every Zelda game, and it ends in a way that there seems to be another adventure in store for Link. Overall, the ending wasn't dark, as most endings aren't.
TP - Like every other Zelda title, the conflict is resolved in TP. Once Ganondorf is defeated, Midna regains her true form, as the curse that was cast on her was released. Then, she returns to the Twilight Realm, shattering the Mirror of Twilight, breaking the only passage between the two worlds. Link returns the Master Sword to its pedestal, and all the children in Ordon Village return home to be with their families. Link, however, rides away from Ordon, possibly embarking on a new adventure. In both MM and TP, you have to say farewell to important characters, including the sidekicks that you had throughout the game. Neither of them are particularly dark in a way, so the final verdict is that the two endings are pretty much equal in said "darkness", as both of them resolve the conflict initiated near the beginning of the game.
Ending - Resolve to Initial Conflict - MM = TP
So, judging by my analysis and the opinions I stated, there are 3 aspects that are darker in Majora's Mask, and only 1 that is darker in TP. Some may argue that those 3 aspects should be grouped together to form a "Gameplay" aspect, but I think those aspects should be individual. The only thing that makes fans think that TP's darkness can be compared to MM's darkness is the graphics, and I think that in pretty much every other part of the games, MM is a darker game than TP.