Generally you will find that a Golf course will not officially close. There will be no opening or closing times. Although players start at sun rise and the last players will simply walk off the course when it is too dark to carry on playing. Pro shops will usually stay open until about 3 hours before sundown.
South Shore Golf Club is one.
Scotland and England
The Masters
There are currnetly 9 golf courses in the British Open rotation, they are The Old Course St. Andrews, Carnoustie, Muirfield, Turnberry, Royal Troon, Royal Birkdale, Royal St Georges, Royal Liverpool and Royal Lytham and St Annes.
Check your local golf course (Municipal or Private clubs open to public). They usually have men's and/or ladies golf club that holds regular tournaments where all members can play. For beginners I suggest finding a par 3 course that has a club. These are easier courses for beginners
Never In Red Deer Alberta Canada there is a little known Pitch and put course open to the public.
Golf courses are an unnatural ecosystem that can be maintained by large amounts of fertilizers, weedkillers and pesticides. All of these chemical can leach into surface or groundwater to cause significant pollution. Golf courses are generally seeded with grasses that do not grow naturally in the area, They must be supported by copious use of water depleting local reservoirs. The open terrain of a golf course usually does not support native birds or insects. The lack of these communities can impact a large area of the surrounding land.
A 'public' golf course does not have "members." The word "public" implies that it is open to anyone (the general public) for play. There is no membership requirement. Only "private" golf courses can restrict their membership rolls.
St. Andrews Scotland is the home of the Royal and ancient golf course. It is one of the oldest golf courses in the world and one of the most respected. It has been the home of the British Open for many years.
Many courses try to stay open until the weather is cold (or snowy) nearly every day. Once the cold weather sets in, many of these courses close for the winter. However, there are some courses that do remain open in one of two forms. Some courses are open if there is a "nice" day. Usually that means a temperature nearl 50 degrees or middle to upper 40's, the greens are not frozen, and the course is not too muddy. Almost always carts will not be permitted because of the damage they may cause to the course in cold weather. Other courses, typically 9-hole courses from my limited observation, remain open in an unofficial sense. They may simply ask you to drop money into a slot and you can play the course (or practice on the course) even though no one is there. Talk to the people who operate the course.
It depends on each specific course's schedule. Most are open every day (in season), only closing for holidays and observances.
March 1 to November fist all week long.