I'm playing as a Swordsage, and my DM is concerned that with Dance of the Spider and the Shadow Jaunt line, it will be impossible for terrain to be a challenge, making it hard to design puzzles. Is this something that he should be concerned will break the game, or is it something that he just needs to plan around? I've been comparing it to the wizard spell Passwall which gets rid of most obstacles.
No, these abilities are no more game breaking than other things characters will have around level 5, the minimum level for Dance of the Spider. Take for example that a wizard could cast Dispel magic at that level, which can render any magical trap harmless, or dimension hop which is a short teleport like shadow jaunt, and they can also cast spider climb. Or a cleric can cast water walking and water breathing, which would negate a lot of the danger of water. And don't get me started on shatter which both can cast
The point is your abilities are meant to help you resolve the conflicts you face, whether it is combat, a trap, or puzzle.
The GM's concern likely comes from the fact that you can use the abilities infinitely. But they have some limitations as well, like shadow jaunt takes a prepared maneuver slot that will reduce your combat ability, and dance of the spider will require you forgo a more combat oriented stance like island of blades.
In my opinion as a GM myself, it is not something that should be considered any more game breaking than any other spell/ability, it is just another tool to solve problems, which may work some times, and may not work other times. The GM should challenge the party by presenting them with a variety of encounters that can't always be solved by one or two go to abilities.
She/he's right, but their view is insufficient.
Terrain puzzles can be fun for low level parties. Unfortunately, with the advent of fly, levitate, or spider climb (etc...) at levels 5 and 3, or with creative use of other magic at similar levels, most puzzles where difficult terrain is an issue... are simply not an issue.
Assuming your DM really likes the environment-as-puzzle, they can certainly create environments where teleporting and other odd movement types are contraindicated by the specific environments. She/he should be aware, however, that most parties will just circumvent these environment unless they buy into the puzzle-as-fun.
If you don't plan on using the abilities to trivialise any sort of puzzle that is designed to challenge the players (as opposed to the characters), as a lot of puzzles are, then let your GM know that.
Personally, speaking as a GM, puzzles can require a sidelining of normal in-game thought processes, the players have to buy into solving the puzzle themselves. Otherwise it becomes a sequence of rolls for clues, effectively a 4e skill challenge. If you make that buy in, then having the ability is not an issue.
If you are talking about tactical combat puzzles, however, then that's a different issue and you something is going to have to give.