More often than not, when someone in the wedding party needs to get “fired,” it's typically a bridesmaid, perhaps because brides expect more emotional and practical help from their attendants than grooms do. However, this isn't always the case and many of the same considerations apply regardless of whether the attendant is "his" or "hers."
Before deciding that a friend has changed for the worse since being invited to join the wedding party, it's always wise to ask if you, the bride or groom, is the one who has changed. Wedding planning tends to follow Parkinson's Law by expanding to fill the time available. And even if the wedding is not the first topic on your tongue every moment, the fact of being half a couple can change relationships with friends. A friend who feels stressed in his work, unsatisfied in his romantic relationships, or just plain jealous of the bride's call on the groom's time can start to act "weird." Don't blame the best man for behaving strangely until the groom has made the effort to spend some "guy time" with him, doing the things they used to do together before the wedding became an issue. It's also worthwhile for the groom to do some probing about whether there's anything wrong in the best man's life that is making him difficult to reach, irritable, or uninterested in the wedding.
Of course, there are always the friends who, in fact, do act weird in ways you've never expected, for reasons that cannot possibly be your doing. Sometimes a wedding brings out a side of a person that was previously easy to ignore. Maybe Jill was always a control freak, but it didn't matter until she started insisting that everything about her wedding was perfect and everything about yours is wrong. Maybe Jack was always flaky about appointments and prone to make stupid comments about how marriage ties a guy down, but until now, these comments were only minor irritants. Now that Jack is expected to provide his measurements for a tux, throw a bachelor party, and generally make himself helpful to the groom, he has lots of opportunities to miss appointments, than make excuses for himself by making cracks about marriage.
Removing someone from the wedding party is a good way to lose a friendship permanently, which is why a "firing" is a last resort for times when the friendship is dead anyway. Usually, it is wiser to aim for a negotiated resignation. The bride or groom (whoever is the injured party) opens negotiations with tactful sympathy, along the lines of "I've noticed you seem distracted and stressed about being in the wedding. You know I want you there, but I'd understand if you were feeling overburdened and having second thoughts. I'd hate to lose you, but I'd feel even worse if you were dreading having to actively participate and you didn't feel you could tell me."
With any luck, the troublesome attendant responds honestly, and the matter is painlessly resolved one way or the other. This method works with bridesmaids who are worried that they'll go into labor during the vows, college friends who'd love to participate but have realized that they don't have any money for the necessary clothes, and pals of all sorts who are delighted with your marriage but are finishing medical internships, studying for the bar exam, working two jobs, or otherwise so stressed that they want to shriek at the mere words "bridal shower" or "bachelor party."
If the troublesome best man is simply avoiding phone calls, emails, and all efforts to contact him, the problem is to decide when to give up! Start with a friendly phone message or email, one that asks a non-wedding-related question that requires an answer, once a week.
If you (the groom) reach four weeks with no response, the message changes to stating that you're worried and want to know if he's okay.
At week six, the message changes to asking if you've done something wrong that you need to talk about.
At week eight, you state that, since you haven't heard from him, you assume he doesn't want to be in the wedding, you feel sad about it, but you have to proceed with plans. Vary the time of day of the contacts, to maximize your chances of actually reaching him. Even a person with a stress-strewn life, a major personal tragedy, or elaborate vacation plans can be reached within eight weeks. If you get no response during that entire time, you have to assume that the attendant has passively resigned.
It happens, occasionally, that an attendant does something so egregious that firing is the only possible solution. The most extreme case I've heard of was the bridesmaid who announced that she didn't even like the bride--which might be no surprise from the groom's sister, but was a shock from the bride's childhood friend. (Insisting on a moderately flattering dress, reasonably comfortable shoes, and one's normal hair color are not, in a bridesmaid, grounds for firing.) In this rare instance, the only thing to say is an icily calm, "I'm so sorry you won't be in the wedding." Then drop it.
One of the most important things to remember is that reaching an agreement that maintains the friendship (if maintaining it is possible) while simultaneously resolving the problems that have led to the groom's frustration, is much more important than "winning" by having the troublesome attendant admit wrong.