While not exactly the same as using Delphi's TIniFile, I've found the open-source Nini library to work well.
Here's an example of reading an INI file using Nini:
; MyApp.ini
[Logging]
File Name = MyApp.log
MessageColumns = 5
MaxFileSize = 40000000000000
Below is a C# example piece of code that describes how to access the configuration data from the INI file from the file above:
using Nini.Config;
IConfigSource source = new IniConfigSource("MyApp.ini");
string fileName = source.Configs["Logging"].Get("File Name");
int columns = source.Configs["Logging"].GetInt("MessageColumns");
long fileSize = source.Configs["Logging"].GetLong("MaxFileSize");