A couple suggestions, because I have to work around this stuff all the time.
1. Try opening the AI file in photoshop. Photoshop will rasterize it more readily than Illustrator, or copy it in Illustrator, switch to Photoshop (Photoshop will rasterize whatever is on the clipboard), make a new file and paste it into that (this may take a while depending on your machine).
2. What resolution are you trying to save it at? I recommend trying 72 dpi. It should still look good, especially since (I assume) people won't be looking at it really close up like they will a magazine.
3. If neither of these solutions work, there is a more complicated fix, and this is one of my special tricks of the trade

. I have used this alot and it seems to work for almost everything. Put each element of your artwork on a layer in Illustrator. Then you can export each element in turn, just make sure you turn off all the layers except the one you are exporting. Open up all your elements in Photoshop. They should all have transparent backgrounds, so you will not need to try to select them out of a white background. Go back to Illustrator and turn on all your layers. Select the paths for all your elements. If you have any photos or special effects on anything, make a simple path for them or Photoshop will try to rasterize them, you will be stuck waiting as it converts them, and then you won't be able to do the next step. Go back to Photoshop and use the paste command. You should get a dialogue box that says "Paste As: Pixels, Paths, Shape Layer". Select "Paths". Then line up your background path with the edges of your background image in your Photoshop file. You can then use all those paths to place the other elements from your other files. Flatten image and voila, your new tiff!
I hope that was clear. If you have any more questions, feel free to sitemail me.