If the file is in the bin directory then it is something that it is referencing.
You can run the SysInternals Process Monitor tool in order to see what the missing file (or permission error) is.
fuglogvw is another useful utility.
When you download the API it delivers a bin and bin64 directory. I am just wondering whether or not you using the correct one. I would assume that you are using 64 bit ASP.NET worker processes?
You did copy all the ESRI files from bin or bin64 into your bin directory, correct?
If you have done all of that correctly then I suggest to open the C++ DLL in the Depends utility (Google it). The thing that I want you to check is whether or not MSVCP110.DLL and MSVCR110.DLL can be found. If not then that is your problem. One think that you didn't say is whether your code is C++ or .NET. If it is C++ and you linked with the debug library then the problem is probably that the server machine does not have the debug CRT.
'Esri.FileGDBAPI.DLL' is the C# wrapper. Do you also copy 'FileGDBAPI.dll'?
Not if I said it works fine in a windows form application, which tells me there is no problem with the dll, is so or not?. The problem is when I try to create a web application or a WCF and references added, throws me the following error: