A "constitutional crisis" is simply a situation which the Constitution does not explicitly cover, usually pitting two branches of government against each other. It's generally resolved by whoever can make the better case for their interpretation, which is very dependent on the details of the situation and the people involved.
As is often the case, FiveThirtyEight has a relevant article. In it, they break the relevant section of the 25th Amendment (Section 4) down into chunks and examine the potential problems with each. Quotes are the Amendment (with my emphasis), text is my summary and elaboration on the article:
Whenever the Vice President and a majority of either the principal officers of the executive departments or of such other body as Congress may by law provide, transmit to the President pro tempore of the Senate and the Speaker of the House of Representatives their written declaration that the President is unable to discharge the powers and duties of his office, the Vice President shall immediately assume the powers and duties of the office as Acting President.
Congress has not explicitly created a body, so it falls on the "principal officers of the executive departments". This is not explicitly the Cabinet, but that's a pretty standard interpretation. No crisis here.
There could be a crisis around what "unable to discharge" actually means, but it's fairly safe to assume that - like "high crimes and misdemeanors" for impeachment - it's up to the body in question. But if the Cabinet's definition isn't broadly accepted, then you have the makings of a crisis.
Likewise, there could be a crisis around Congress' ability to create a new body that can declare the President unable - more on this later.
Thereafter, when the President transmits to the President pro tempore of the Senate and the Speaker of the House of Representatives his written declaration that no inability exists, he shall resume the powers and duties of his office unless the Vice President and a majority of either the principal officers of the executive department or of such other body as Congress may by law provide, transmit within four days to the President pro tempore of the Senate and the Speaker of the House of Representatives their written declaration that the President is unable to discharge the powers and duties of his office.
There's one key hole in this section: The use of the word "unless" instead of "until". The President resumes the office unless the Cabinet does something they have four days to do. Is he President during those four days? Does he have any authority? Can he fire the entire Cabinet during that time? What if they don't act during that time?
So here's one potential Constitutional crisis: The President declares that he is fit, and fires the Cabinet members that signed onto removing him. They counter by saying that he doesn't have authority to fire them, since they haven't yet run out of time to respond. Crisis.
Bonus crisis if the non-fired portion of the Cabinet explicitly says "No, we think the President is fine". Now you have two potential bodies that can notify Congress of the continued inability of the President - the VP's Cabinet and the President's Cabinet. Does it take a majority of either half (with them assuming that the other seats are now vacant), or a majority of the whole thing?
If the Congress, within twenty-one days after receipt of the latter written declaration, or, if Congress is not in session, within twenty-one days after Congress is required to assemble, determines by two-thirds vote of both Houses that the President is unable to discharge the powers and duties of his office, the Vice President shall continue to discharge the same as Acting President; otherwise, the President shall resume the powers and duties of his office.
This has the same as before, except worse. Now there's up to twenty-one days where the President's status is in limbo. Who's running the country during that time? What if the VP-as-Acting-President takes actions which the President can't undo if he resumes office (pardons, firings, signing or vetoing legislation, etc)?
Additionally, since it takes a 2/3 majority of both houses to override a veto, and a 2/3 majority to create a body that can declare the president unable to carry out his office (as previously mentioned), then the supermajority of Congress and the VP can conspire together to bypass the rest of this amendment and force the President out. As before, if this is not seen as legitimate, crisis. (Even though the bar is higher to do this than for an impeachment, it doesn't require the excuse of a "high crime [or] misdemeanor" to remove an "unfit"-but-not-"criminal" President.)
None of this is to say that a crisis will happen if Section 4 of the 25th Amendment is invoked. Just that there's a lot of potential ambiguity where a conflict of interpretations will become one.