The answer is that it depends where you go. Not all Muslim-majority countries are the same. Indonesia, for example, which is the country with the largest population of Muslims on the planet is generally a far more tolerant and relaxed place to live than many Gulf states. This is partly to do with the fact that there are many other religious denominations in Indonesia and also that indigenous belief systems are still held in high regard.
Whichever Muslim-majority country you decide to teach in you will have to be aware of the holy month of Ramadan. All healthy Muslims are supposed to fast during daylight hours so you will be expected to eat and drink only in private. Even global food chains such as Dunkin Donuts put up curtains so that non-Muslims can eat inside and not ‘tempt’ Muslims outside to do the same. It is quite an eye-opener to those not accustomed to it.
The real issue to watch out for is Sharia Law. Although this law should obviously really only apply to Muslims, some governments and regions of certain countries have tried to make it the official law for everyone, regardless of your religion. Brunei is currently implementing this, and many English teachers who had considered Brunei will most likely be rather concerned. The same can be said for the semi-autonomous province of Aceh in Indonesia.
In neighbouring Malaysia, Sharia Law only applies to Muslims. So, when you are shopping you will see that it is illegal for Muslims to buy or consume alcohol. If you are not Muslim then you do not need to worry about drinking alcohol, except if you are in the company of Muslim people who might be offended by your actions.
Not only is there a potential problem with alcohol. There is also a potential problem with pork. Eating meat from a pig is haram (sinful) for Muslims so it is a good idea not to eat pork publicly in Muslim-majority countries unless you are in, for example, a Christian-owned restaurant where it is common to do so.
If you spend any length of time in a Muslim-majority country you will become aware of the call to prayer which happens several times per day in local mosques. Sometimes it can be quite loud, as microphones and speakers are set up in order to broadcast it over a wide area. This can come as quite a shock, especially the pre-dawn call to prayer which can be as early as 4.30am. Choose your neighbourhood carefully.
Generally speaking though, you will find that people of whatever faith are very welcoming of strangers. The key to successful relationships between those of different religions is mutual respect. If you respect their beliefs they will surely respect yours too.