I live in an Islamic country, but I don't want to be a Muslim. I have no choice but to claim to be a Muslim: If I leave Islam, they will execute me according to Muslim law.
Where can I go to have a peaceful, non-religious life? Can I get asylum in any English-speaking countries?
For example, would the Republic of Ireland accept me as a refugee?
This is a hypothetical/abstract question of law. I am not making any personal statements about my beliefs or intentions.
Legally speaking, very many nations grant asylum, and religious persecution is one of the most basic grounds for granting asylum, following the 1951 Refugee Convention. This newspaper article compares asylum statistics in Ireland versus other parts of Europe. The Irish immigration authorities spell out the details for an asylum application. Note that you must already be in Ireland, to apply for asylum in Ireland (you should apply when you enter the country). One could also apply to Norway (almost an English-speaking country), but again you have to be in Norway to do so. There is a generic solution to the "what if I'm not in country" problem via the UNHCR, which can propose resettlement into various countries.
I need to add that getting a visitor's visa from certain countries can be extremely difficult. To take an example, Norway (which is fairly open to refugees) is pretty up-front on the chances of getting a visa, based on country. To take a random example, they are not very optimistic about visitor's visas from Iran, and they say "we consider how probable it is that you will return to your home country or the country you live in when the visit is over. We consider the situation in your country and your own situation", "If we believe that it is unlikely that you will return, your application will normally be rejected" and "If you plan to visit Norway as a tourist, you will normally not be granted a visa". This is the fundamental problem that refugees face, the problem of getting there.
One country that allows visa-free admission from Iran is Turkey. This guide (which is in Farsi so I can't comment on) provides practical information on the UNHCR asylum process "the political asylum process for Iranians in Turkey": that may indicate that apostasy is a different matter. Other evidence suggests that this option is worse than staying put.
Only for the sake of discussion, Svalbard is a theoretical possibility. Svalbard (next to the North Pole) is part of Norway, but Norway treats it as being somewhat outside of Norway. It is outside the Schengen visa area, and it is a visa-free zone, meaning that nobody requires a visa to visit or live there. This is due to the Svalbard Treaty whereby Norwegians and treaty nationals have equal rights to the islands, and while most nations are not treaty signatories, it has been policy to extend those rights to everybody. The Governor does have the power to expel anyone who is a burden on local society (e.g. unemployable). Normally one would have to get a Schengen area visa to get there, which would be an obstacle, but it is apparently possible to get a same-day visa-free transit at Oslo Airport, if travelling non-stop to Svalbard (I cannot find a definitive policy statement on this matter, but I also am not sure where exactly to look). There are some air routes from outside Schengen where the first Schengen stop is Oslo. The Governor's office gives appropriate warnings about local problems (ridiculous prices, housing shortage, work shortage, more polar bears than people, really cold).
You should contact an organisation specialised in this area. Here is one for the UK, with specific information on asylum seeking apostates:
https://www.ex-muslim.org.uk/2010/09/apostasy-and-asylum-in-the-united-kingdom/
You are asking here on the law stack exchange, but law is only a small part of the equation. Actually getting to a point where you can apply for asylum will be more difficult than the legal question, and then ensuring that the asylum request is accepted is again a matter that only on the surface is a purely legal question. An organisation of ex-muslims like the one linked above (there are others in other countries) will help with the entire process.
