If you're in the same situation I'm in, you've just spent days searching every variation of "latest flash player doesn't work with IceWeasel or Firefox in Debian Squeeze", including this thread.
There are a number of issues concerning use of the latest Flash Player in IceWeasel and Firefox.
IMHO, IceWeasel is outdated and should be replaced with the latest version of Firefox. Installing the latest version of Firefox can be difficult, but is well worth the effort.
Please note: If you have a 64 bit CPU, make sure you install the 64 bit versions of Firefox and Flash.
Starting out by installing the latest version of Firefox is a good first step. Removing any existing versions of Gnash and Flash is a good second step. Installing the latest version of Flash is a good third step. But this may not solve your problem.
There are problems with using the latest flash plug-in with older CPU's. The latest Flash requires CPUs with the SSE2 instruction set. Older CPUs may not have it. If you have an older CPU don't despair there is a fix. Use the following command in a terminal to discover if your CPU has SSE2:
grep flags /proc/cpuinfo|sed 's/ /\n/g'|grep sse2||echo no SSE2
(don't forget to use the shift key when copying into a terminal, i.e. ctrl/shift/v)
If your CPU doesn't have SSE2 then you're going to need an older version of flash. I found version 188.8.131.52 is the newest version that still works with CPU's that don't have the SSE2 instruction set. Download the RPM at:
(this was the only sight where I could find this flash plug-in)
Open the rpm file with an archiver (you should be able to double click it in Debian Squeeze) and drill down until you find the libflashplayer.so file (should be 16.3 meg), select it and extract it to your Desktop.
Copy the libflashplayer.so file from your Desktop to /usr/lib/mozilla/plugins, or wherever you installed the newer version of Flash that doesn't work. If you're not sure where it is, do a file search for libflashplayer.so and replace all the instances you find with the version you just extracted (sorry about the shotgun approach, but at this point you're frustrated and angry and just want the stinking thing to work. You can clean it all up after it's working). Fire up Firefox and see if it works. If not, you've probably done something wrong when you tried to install the latest version of Flash. Do a web search for installing Flash in Debian Squeeze. Just make sure you use the libflashplayer.so file you just downloaded. DO NOT follow the instructions to download the latest version of Adobe Flash.
Again, this fix is for old CPU's that don't have the SSE2 instruction set. Firefox combined with a more current version of Flash is amazingly better than IceWeasel with Gnash!
Now if Adobe will add the simple code necessary to allow it to determine which version it should fire up, we'll all be better off. Nevertheless, thank you Adobe for this awesome piece of FREE software!