I recently upgraded my fixed gear bike to an odd gear ratio and now the chain

If you have horizontal dropouts there should be plenty of room- at least 1" (25mm) of adjustment which would easily allow for proper chain length, so I am assuming that you are a little confused between horizontal (sliding) dropoutsd and vertical ones. This is pretty common, actually.

Anyway, if you do indeed have vertical dropouts that don't allow you to slide the wheel back then yes, you need a chain tensioner.