Most likely: MC is writing one or more fields to the music file each time it plays, which causes the file date to be updated by the operating system due to the write.
To see which fields appear as tags in the file, select a file and then expand the section Tag Dump, listed in the left pane Tag window. The tags appear a little differently, depending on the type of music file. For example, in a flac file they appear in the Vorbis Comment Block. Look for tags that contain date last played, number of plays, etc. that change with each play.
Once you identify the tag(s) that change, find the corresponding field in Options > Library & Folders > Manage library fields... and uncheck the box [ ] Save in file tags (when possible).
[Note: You can get a list of all fields which MC has marked for saving in tags, but it probably includes many fields which you are not using and hence do not actually get written as tags: Options > Library & Folders > Manage library fields > Show only fields saved in tags]
Alternatively, use your favorite binary file compare utility and examine the file contents of the old and new files side by side to explicitly see the differences. All tags appear in the "header" section, preceding any embedded images and music data.
FYI, my favorite compare utility is Beyond Compare. Its new Version 5 includes a media compare mode (in addition to hex compare), which has a metadata section that beautifully lists the tags in two flac files side by side (either all or just differences), fully human readable. The metadata list even compares hex content of embedded pictures!