Why doesn't Alan Shore contest for partnership?

Basically because he'll never make it. Throughout the series it's shown that he's a 'loose cannon' and, if made a partner he will basically bring the whole firm into disrepute.

He has resigned himself to the fact that he will never be made partner at the firm due to his unpredictable behavior and lack of trustworthiness. While the managing partners of Crane, Poole & Schmidt do not fully trust Alan, they do recognize his talent as an attorney and will often give him the latitude to go about his unorthodox means if they feel it will benefit the firm.

Wikipedia

Alan's position at Crane, Poole, and Schmidt is technically that of an associate, yet for all practical purposes Alan is treated as a Partner in the firm. Alan himself has acknowledged he has no hope of actually ever making Partner, for the very good reason that the partners, by and large, do not trust him, yet he is treated as much more of a peer by the named partners and senior partners than a subordinate. He generally exerts influence with both the Senior and Named partners far above what his actual position would indicate, including his strong friendship with Denny Crane and an often grudging respect from Schmidt, as well as often ignoring the rules and warnings of serious consequences, if he feels strongly about something.

Bostonlegal.wikia.com

I hope I'm not too late to add.

So in most law firms in countries that follow the American and British system of partnerships, a partner must buy a stake into the firm. Now that they have some skin in the game, if the firm fails then they lose money as well. Partners don't really receive, they get paid via equity from the firm. Essentially instead of paying a high tax that comes with a salary, they pay Capital Gains tax that comes with equity and that tax is very low. However the if one partner is doing illegal activities then in the eyes of the court it will be assumed that ALL the partners had some level of responsibility. As partners you need to hold each other to a high standard so only people you trust, people that you know will tow the line will be made partner.

In Boston Legal it shows that Alan doesn't always tow the line and that he is a loose canon.

In The Practice (The final two seasons can be seen as a prequel to Boston Legal), Alan doesn't confirm nor deny that he was embezzling money from one of his clients. He pretended to be an airline executive to help prevent a young lawyer from being disbarred.

Also in The Practice we see that Alan realises that there is more to life than making partner and having lots of money (check the episode where he helps a homeless dad and his daughter), he realises that his true passion is to fight for those that can't fight for themselves and if that means breaking the law and losing the trust of your colleagues then so be it.

There are two reasons: the one Alan Shore presents himself, and non-obvious one.

The former is that he is a loose-cannon, in Season 2, Ep 11 he retorts to Shirley, that he is not a partner, [I'm not a partner] but only because I can't be trusted.

The not so obvious reason is that it would contradict the essence of this hedonistic character, that James Spader plays on the show. One of the primary responsibilities of the partner is to bring new business, shmooze with clients and supervise other cases. None of the activities that Alan Shore enjoys, especially since as a partner he would have to tone down his antics.

Being an ultimate hedonist, why would he do something, that wouldn't bring him pleasure?