Well, I feel that while nobody should be forced to carry out a procedure or provide a service which is against their conscientiously held beliefs, equally nobody should be permitted to prevent or obstruct a person from exercising their legal rights.
In Britain, the Abortion Act 1967 carries a conscientious objection clause, allowing doctors to refuse to participate in terminations but obliging them to provide emergency treatment when a woman's life may be jeopardised. The BMA (British Medical Association, the professional body for doctors) advises that doctors who feel unable to participate in abortions still have an ethical duty to refer patients to another colleague and that preliminary procedures such as clerking in the patient and typing referral letters (as in a test case) are 'incidental to the termination' and are to be considered outside the scope of the clause.
There have been cases of NHS interview panels asking candidates whether their beliefs would prevent them from carrying out incidental and non-incidental procedures to do with elective terminations, and of refusing to appoint based solely on answers given to such questions. This is deemed "inappropriate" and in 1994 the NHS Executive issued guidance to all English and Welsh hospital trusts, instructing them not to question candidates about their personal views on abortion.
However, although health is a devolved matter in Scotland, abortion is reserved to Westminster, and the NHS guidance has never been issued in Scotland. Scottish trusts have stated that they feel it is "reasonable" to ask such questions.
As for gay marriages, I guess I feel that there would probably need to be opt-out clauses for priests who felt unable to perform a ceremony. In the Uk, only men and women can be married, so the question does not arise.
Many doctors have known since time immemorial that the Hippocratic Oath carries a hidden subclause "... thou shalt not strive officiously to keep alive", and while I am not saying there are dozens of Shipmans practising, there have been some odd cases, notably the Eastbourne one in the 1950s, involving Dr John Bodkin Adams. (Look it up in Wikipedia).