How best to read a File into List<string>

var logFile = File.ReadAllLines(LOG_PATH);
var logList = new List<string>(logFile);

Since logFile is an array, you can pass it to the List<T> constructor. This eliminates unnecessary overhead when iterating over the array, or using other IO classes.

Actual constructor implementation:

public List(IEnumerable<T> collection)
{
        ...
        ICollection<T> c = collection as ICollection<T>;
        if( c != null) {
            int count = c.Count;
            if (count == 0)
            {
                _items = _emptyArray;
            }
            else {
                _items = new T[count];
                c.CopyTo(_items, 0);
                _size = count;
            }
        }   
        ...
} 

A little update to Evan Mulawski answer to make it shorter

List<string> allLinesText = File.ReadAllLines(fileName).ToList()

Why not use a generator instead?

private IEnumerable<string> ReadLogLines(string logPath) {
    using(StreamReader reader = File.OpenText(logPath)) {
        string line = "";
        while((line = reader.ReadLine()) != null) {
            yield return line;
        }
    }
}

Then you can use it like you would use the list:

var logFile = ReadLogLines(LOG_PATH);
foreach(var s in logFile) {
    // Do whatever you need
}

Of course, if you need to have a List<string>, then you will need to keep the entire file contents in memory. There's really no way around that.