I created an account just to get into this discussion, because I have recently seen the ending of Halo 4 (still sad Cortana died [assumed]).
Here's my guess of why they remove his armor IN FRONT OF SPARTANS*
It's probably some psychological or mental relief or stability. I know that the Spartans are made to not feel MUCH emotions as an average human being, but the fact that Chief took off his armor in front of all of the other Spartans shows or can resemble that his character and development, are unfolding/unraveling. Plus, I'm pretty sure we all know Chief has been building up a loveness or relationship to Cortana, and seeing her fade away probably made Chief realize something important (he also re-stated the quote he heard from Cortana that is something about machines). He's turning more into an average human (mentally of course) and developing more emotions.Cortana's pass-away brought a more, emotional and human side of Chief too.
That's my guess.
Oh and, like the others said:
He probably doesn't care about getting his armor repaired, changed, modified in front of the Spartans because he probably already been through this routine multiple times. Also, his character is unraveling as previously stated.
