If you were conscripted into the army but refused to kill, what happened to you?

I'll only answer for the First World War and New Zealand. Initially the requirements of the New Zealand Expeditionary Force were able to be filled by volunteers, however, as the NZEF was expanded, and the rate of volunteers slowed, authorities began considering conscription. In late 1915 the government carried out a 'national registration' to find out how many men of military age there were in New Zealand. Of the 196,000 eligible 33,700 indicated that they did not want to serve in any capacity, and a further 44,300 were not willing to serve overseas. Now there would be a lot of reasons for people to object to joining the army, but this group would have contained many men who objected on principle. The government said that this registration would not be used as a basis for conscription, but, in August 1916, it (fairly inevitably) was (film of the first ballot being drawn). If called up, exemptions could be applied for, but were rarely granted, and, when they were, were mostly on grounds of excessive hardship, or being indispensable to their employer (for an example of the level of sympathy usually displayed by these boards, see this article. John Linkster appealed on grounds that two of his brothers had already been killed and the third was already serving, "the Chairman: Where is the hardship to you? You have no parents, and what is to prevent you doing your duty as your brothers did before you?").

Automatic exemptions were only granted to members of a church 'the tenets and doctrines of which declare the bearing of arms and the performance of combatant service to be contrary to divine revelation', though they were expected to perform non-military service in New Zealand. This meant that a considerable number of men opposed to the war were not exempted. When called up, many of them were offered service in the Medical Corps, or another non-combatant role, but almost 300 men, who refused this or were not offered it, were imprisoned. They were a mix of religious objectors who were not part of a pacifist denomination (or were part of a small unrecognised one), socialists, Irish nationalists, Waikato Māori, or objected for other reasons. Almost all were sentenced to hard labour. New Zealand's socialist paper, the Maoriland Worker, ran regular articles on them and appeals for their families.

In 1917 14 of the most stubborn objectors, including three of the Baxter brothers were sent to the front, where they were subject to considerable physical brutality to try to convince them to give up. Eventually one was recognised as a genuine objector under the Act, the others agreed to serve as stretcher-bearers (one would subsequently be killed in action), Mark Briggs, who refused to walk towards the front and had to be dragged, was injured so seriously that he had to be discharged, and eventually the final objector, Archibald Baxter (father of the poet James K Baxter) was also hospitalised, where English doctors declared him insane, and unfit for service (his book about the experience: 'We Will Not Cease' can be read here).

I briefly mentioned Māori above, to expand a bit: Māori were initially exempted from conscription, in 1917 conscription was extended to Māori in the Waikato-Maniapoto area, where volunteering had been poor. This was largely due to the rather dim view many of them held of the government, particularly among Waikato, for whom the massive land confiscations that had followed the New Zealand wars of 50 years earlier were still a very fresh grievance. Few of the men conscripted turned up. A police raid arrested 14 conscripted men, but very few conscripted Māori men served, and the war ended before any were sent overseas.

sources: the penguin history of New Zealand by Michael King, Te Ara encyclopedia, and a bunch of pages from the 'New Zealand History' website, run by the Ministry for Culture and Heritage. (This is the best overview of conscientious objection)