Anyone who wants to can post a response to some or all of the points - please make it as clear as possible which you're responding to. Hopefully this will effectively become a multi-author FAQ.
General objections to consequentialism
1) In some situations [insert your favourite thought experiment here], consequentialism says we should do something which is clearly immoral.
2) In some situations, applying consequentialist ethics would create a situation which is clearly undesirable.
3) In still other situations, making consequentialist deliberation would lead to negative consequences
4) In yet other situations, applying consequentialism would mean becoming non-consequentialists.
5) Most consequentialist values are impossible to measure, so following aggregative consequentialism is impossible.
6) It’s impossible/undesirable to impose a structure on ethics. Better to rely on our judgement, and take one situation at a time.
7) Saying we should prefer some type of consequence is like saying we should respect people’s rights – a fundamental axiom that you either accept or don’t.
8) It's very hard or impossible to evaluate actions the way consequentialists say we should, because they might have very long term consequences.
9) I don't like applying labels to myself.
Specific objections to utilitarianism
10) Happiness isn’t a universal sum which everyone contributes to or subtracts from, it’s a subjective experience that varies from each individual to the next. Utilitarianism does not take seriously the distinction between persons.
11) Utilitarianism demands impossible sacrifice from us (this Demandingness Objection seems to apply to most forms of consequentialism, but maybe less so than to util).
12) ) Utilitarianism says that our only goal is to maximise happiness, but people obviously don’t act as though it is.
13) I agree with the sentiment of increasing wellbeing, but I disagree with the idea of using people for the benefit of others. It treats them as less than human because it disregards their intents and goals. So utilitarianism is a good starting point for ethics, but we must also respect human dignity.
Let me know if you can think of any that I've missed, or can think a better way of phrasing anything above. Also, let's keep this thread for immediate responses to the common objections so that it remains clear - if you want to argue a point, better to start it in a new thread.