There are a few reason for this possibly happening.
1. Original File is a template. File should open as a Untitled.indd if it is.
2. The file has read only access to it.
What you say doesn't make sense... If the file is still open there is no problem?
How can you do a save or a save as if it isn't open?
Apart from that the two problems that McKay says might be it or you may be opening the file in a newer version of InDesign than it was last saved in.
Is it showing any missing plugin while opening a document? If yes, it will ask you to save every time you open that document.
Export into .inx format and check.
Working in Indesign CS5. File is CS5, no problem (not a template or a previous CS2 or 4 etc) file named always figures (ie 00000.ind).
The file can only just have been closed and then reopened, because I may have to a further correction. Opening it straight away to do the correction I have to Save As, I can not just Save
Ok is the file on your computer or on a networked shared drive, if on a network drive you probably don't have rights to alter the file and indesign will ask you to save as. Check your log in to the networked drive and that you have permissions to change things on that drive