If you are editing tif files, then you have only a couple of choices.
Flatten the image before you save it. This will reduce it down to one layer. That's the smallest you can get.
When you first open the image, look to see if it's 8 bit or 16 bit tif. It won't help you, but if the image is 8 bit, and you save it as 16 bit, you will have dramatically increased the size of the file for nothing. If it's started as 16, no worries!
check if you are saving with compression. Tif compression is lossless, so unless you did something like resize+resample, it remains the same bit density, etc, so should be just as printable as what they sent you (unless you messed up!).
Cheers!
