Q:

How to deal with an incompetent colleague? 

The setup...
A little more than a year ago, we started a search for a "rock star" developer.   We had lost our legitimately brilliant lead developer to a start up.   I was able to step in and fill his shoes, but we needed a really strong person to come in and backfill for me.   
We searched for a few months but couldn't find anyone.   Finally, our department director "found" a guy.   They were clearly friends before.   The candidate (we'll call him Roger) was interviewed.   Roger had no experience in C# but had done some programming in some strange system somewhere.   He failed the technical interview pretty spectacularly.   None of us developers thought he was able to do the job.   Despite the protests from the rest of the team, Roger was hired.   
The problem
In the time since Roger has been hired, he has picked up C# and web development.   He has the skills of an intern or other very entry level person.   There's no doubt he's not the rock star we need.   Even Roger acknowledges that he's in over his head.   He really wants to be better and in time, he might be able to step it up. 
The problem is that he thinks that because he's been out of college for 10 years or because he's been friends with the director he has some say or sway over us.   I point out to him the things he's doing wrong or that he should be doing differently.   He then takes a really condescending tone and gets sort of bitchy. 
The rest of the team has talked to the director and expressed our problems.   We're answered with "he really wants to learn" or "I need you to help him" or "he has skills that we need".   I don't care if he really wants to learn.   I want to be a fighter jet pilot.   Too bad for me.   I don't want to help him when he rejects my help and gets bitchy.   What skills are you referring to?   The skill of working a solid 4 hours a day and pissing people off? 
In short, he has become my problem and I'm not going to get any help from management. 
What I need
Since it's obvious that Roger isn't going anywhere, how can I get over myself and deal with him?   I'm not trying to simply vent as the rest of the team here is sympathetic with me and has the same view.   I really need some strategies I can use to co-exist with this guy. 

A:

If your director won't do anything, and there's nobody else to appeal to, here are the steps I would recommend:
1) If your team agrees, make it a point not to work extra time (or put in an inappropriate amount of extra effort) to make up for his deficiencies.  Money talks loud, and if your director and his superiors see things slipping and ask you about it, you can humbly explain that you attempted to warn them about this man's abilities, and that quality is suffering because of his shortcomings. 
2) If money doesn't talk (it's actually possible), start looking for another job.  I hate to give that advice so early and so readily, but if you have management that is willing to pile on people that are doing the job and let somebody else get a free ride, the company is doomed to fail, now or later, and things aren't likely to get better for you. 

A:

When your director answers with "he really wants to learn" or "I need you to help him", you need to say OK - we can do that, it would take about X amount of hours per week for Y years, and cause our project to be delayed by Z. If the director is happy with that, carry on - try not to vent your frustration on the Roger, and treat him like an intern (if that is his skill level and he needs a large amount of training) - note that this does NOT mean to treat him badly in any way. 
"I point out to him the things he's doing wrong or that he should be doing differently.  He then takes a really condescending tone and gets sort of bitchy." -  instead of pointing things out, offer to show alternatives - or if you aren't the mentoring type, find someone else to do it - in either case, if he truly is "bitchy" about it, that conflicts with what the director says about him "really wanting to learn", and you may need to have a discussion with him and the director about that as a separate issue.  

A:

When a manager is technically incompetent, you can do one of 3 things. 
1) Help Roger become competent - Mentor him, assign in dev task, and hope he will step up to the plate and improve.   This will eat into at least 1 developers time significantly.  
2) Elect a Vice-Project Manager - Have a competent developer step up to a managerial role, and let Roger focus 100% on keeping the project funded and the upper managers happy.   This may have been why he was hired.   See how effective he is at going to bat for the team when there is slippage or unimplemented features.   Don't discount him because he's not a technical whiz-kid. 
3) Find a new job - If you believe that Roger is truly incompetent, and that it will cause you to work nights and weekends, perhaps its time to switch companies.   If you continue to 2nd guessing everything he does, you'll end up burned out and angry at him.