What I refer to, is a scenario where the character has a set of abilities (let's say flame generation/manipulation), and then for puzzle solving, you're presented with obstacles that require external methods to solve. Like a gate or some other path obstruction. Clearly the character could burn it to the ground with little effort, but instead, the character stresses the importance of finding the key to advance. This doesn't even touch on simply climbing over it, etc.

Now, I could see the character being a non-destructive type, but in this example, the character is the destructive type and has already melted down several enemies and miscellaneous debris that sits around the level. So I'm wondering if this is the right way to do things, or is it just terrible game design?

I'm working on a game, and I want to have problem-solving suitable to the character's abilities, but also taking into account how some players may perceive it. If the character comes off as an unstoppable juggernaut, I don''t want him to turn pacifist over a locked door.

Having destructable doors is not unreasonable, nor would it be unreasonable to construct doors that are less destructable knowing that some people out there have destructable talents. In real life we have doors that range from flimsy, "wouldn't stop a fly" affairs up to 20ft thick doors designed to stop the effects of nuclear explosions getting through. Why would you not have a similar thinking and methodology in your own game world? At the end of the day it is not unreasonable to have security on some doors which act as barriers between a less controlled environment i.e. outside and a more controlled environment i.e. inside, other doors used for internal traffic might be more flimsy persay.

Hope this helps

Perhaps as you indicate the key is finding out how to apply the powers, or having good reasons they won't work. For example, can't use flame because you're on an iceberg and it will melt. Or if you have a power like Juggernaut you need to find the way to get a big enough run-up for your powers to kick in.

Those make a lot of sense.

On that note, in games that actually function like I described, is it bad design?

Bad design will break immersion or cause you to sit outside the game and in thinking about it, decide that it doesn't make sense. If you play a game and these thoughts don't pop into your head then (while the game may itself not be great) the design is probably adequate. In designing or writing I tend to work on the rule that if I can't justify it ingame or in the plot then I had better make sure I do so or alternatively dispense with the particular issue. However there are times when you may have to include elements that do not strictly get justified by design or writing that is when you have to become more creative ...in order to obscure or distract a player from losing immersion or sitting back going "this is wrong". Unless you choose to utilise very openly such devices as an "Act of God" or "Breaking the Fourth Wall".

Hope this helps