In this game... http://www.wargear.net/games/view/573701 I did all my attacks and all my fortifies first, then was going to place 5 units to get my reserve down to 15 (the limit).  I placed my 5 units and hit apply.  I think since I had no attacks or fortifies left, it grayed everything out (including place) and thinks my turn is over, but the end turn button never clicked so it still shows me on the clock.  

Not too big a deal in that when the 24 hours goes past I'll just get skipped, but still not great either.

Try the other player and see if it can be ended from there?

I tried both.  Neither works.

It happened again and this time I thought I even kind of went in the normal order of things.  I did my attacks, then placed, then did fortifies.  And it still grayed out the end turn button.  Could it be a problem with the map?  

I've played that board and never had that problem before. The other players in the game don't look like they are having that problem either. Have you tried a different browser?  Have you placed all of your units or are you past your 13 max reserve?

Another possibility - Clear your cache.  Go to your player page (click on your user name in the top bar), then scroll down to the bottom of the first box and click the link to "Clear Cache"