The very first Mega Man 1 game has one of these: in Elec Man's stage, there is an item called the Magnet Beam hidden behind some blocks, which require you to have beaten Guts Man before you can remove them. A level near the end of the game is impossible to complete without this item, and there's no way to go back and get it. Even better, you have to have one of two specific weapons to get the item; if you go through the stages in the wrong order, then you have to visit this stage (and fight its boss) twice. One of these weapons is Elec Man's weapon.
In 2, if you reach the final boss without having enough Bubble Lead (BubbleMan's weapon) energy (from using it all up against earlier enemies in Wily's fortress), fail to connect said weapon with the boss too many times, or were foolish enough to use any other weapon (which restores the boss' health!), then it is impossible to win the last fight. That's not all: when you start your next life, the checkpoint where the game dumps you at has no enemies; therefore, it is impossible to recharge the weapon required to win the fight. Result? Welcome to the game's Hopeless Boss Fight, where you'll have to suicide yourself for the number of lives needed for Game Over. If you reset the game in rage, then you did write down that password to get back into Wily's fortress, right? Oh, and it only puts you back at the beginning of the fortress. To negate this problem, the player can go for the Game Over and then continue; you get all your weapon energy back and start at the beginning of that level instead of the whole fortress.
Of course, if you're using "Invincibility" via Game Genie, then you cannot die and thus have to restart. At least it is noted in the code booklet.
In 3, while revisiting Gemini Man's stage, if you were to drain all weapon energy for Rush Coil and Rush Jet early on, you would come to a wall that's impossible to scale immediately after the first Doc Robot battle. Unless you can cheat using the second controller to clear it, you'd be forced to reset the game.
Mega Man X
In Mega Man X 6, the Amazon Area has a portal to an alternate exit. This area requires the air dash ability to cross a bottomless pit. If you reach this area with X's Shadow Armor, which is incapable of air dashing, then you're screwed.
The same can be said for Ground Scaravich's Stage. The gimmick of the stage is having a random room every teleport you have to go through. Sometimes, one of the rooms contains a jump that requires Air-dash. If you are wearing the Shadow Armor, or if you're crazy enough, no Armor, you really don't have much choice but to die, and hope the stage doesn't teleport you to that room.
The same room can also spit out a gap that requires the Mach Dash ability from the Blade Armor to cross. What's that? You don't have it yet because the fourth part is later on in this level? Too bad!
Unless you also happen to have a specific combination of upgrades equipped (Hyper Dash and Speedster) and a specific one NOT equipped (Jumper), meaning you can get through if you know about this beforehand. Oh, and the necessary parts can be Lost Forever if you fail to save the reploid holding them.
There's a jump almost exactly like this in the final stages. The same situation can arise, requiring either an air-dash or the parts listed above. The penalty for being unable to perform this jump is much more severe: you'll have to replay quite a few stages and bosses.
A similar problem arises if you're trying to get the Dr. Light capsule in Metal Sharkplayer's stage. There is a large chasm you have to jump that requires a certain combination of upgrades, and possibly Blizzard Wolffang's weapon. If you don't have those, then you can't make the jump. Die, and you respawn directly in front of that chasm with no way to get back to the stage's main path. And if you kill your remaining lives off to get a game over, don't hit continue, or else you'll once again respawn in front of the chasm with a full allotment of lives.
And for a 6th example, just from X6: It is possible to get locked into a path where you must fight High-Max without having any special weapons available. High-Max is invulnerable if you don't have at least one to use. Your only option, as in the above cases, is to get a game over and return to the stage select screen. Did we mention X6 is generally considered to be a Franchise Killer?
Mega Man Star Force
In the first Mega Man Star Force game, it was possible to get stuck between an NPC and a wall. If you did this while not on a Wave Road, then the only way to get out was to reset and load from your last save; hopefully, you knew better than to save while stuck. Later games fixed this problem by briefly allowing you to run through NPCs if you get stuck.