Section, segment, or part all have similar meanings, but the situations in which each can be used is different.
According to Oxford Dictionaries Online, a section is
- Any of the more or less distinct parts into which something is or may be divided or from which it is made up
- A distinct group within a larger body of people or things
Therefore, you only use section when the larger thing that the section is coming from can be, or is, divided into distinct parts.
According to Oxford Dictionaries Online, a segment is
- Each of the parts into which something is or may be divided
- Geometry A part of a figure cut off by a line or plane intersecting it, in particular
Thus, a segment is almost exactly the same as a section, and the two can be used interchangeably. However, segment carries a secondary meaning and so can also be used when specifically talking about lines and planes in geometry.
When you look up the word part in Oxford Dictionaries Online, you'll find that it has no less than 15 different meanings and submeanings. It is therefore the most broad and flexible of the three words we're studying, but it's general meaning is
An amount or section which, when combined with others, makes up the whole of something
So you can use part whenever you want to talk about any piece of a larger entity.
In terms of your examples...
"That segment/section of the road is still closed." Segment and section are interchangeable here. In my experience, section would be the most commonly used, but segment is equally valid, especially since a road is more or less a line, and in geometry we have line segments. You can also use part, but you lose the specificity you would have had with the other choices.
"The library has a large biology section." This is a special case. Section is the only acceptable word to use here. The biology section is a distinct part of the larger library. I have only ever heard people using section to reference specific areas of libraries, never segments, or even parts. And I've been in a lot of libraries.
"The tail section of the plane." Section is the most common word to use in this case. You can also say part, if you want to give a more informal tone.